{"id":29596,"date":"2022-05-06T16:54:26","date_gmt":"2022-05-06T11:24:26","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=29596"},"modified":"2023-04-22T20:59:51","modified_gmt":"2023-04-22T15:29:51","slug":"definite-integrals-excercise-problems-with-solutions","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=29596","title":{"rendered":"DEFINITE INTEGRALS (Excercise Problems with Solutions)"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\underline{PART\\ -\\ A}\\]<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ \\color {red}{Evaluate\\ :\\ \\int_1^2 (x\\ +\\ x^2)\\ dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int_1^2 ( x + x^2)\\ dx = (\\frac{x^2}{2} +\\frac{x^3}{3}) \\Biggr]_{1}^{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= [(\\frac{2^2 }{2}+\\frac{2^3}{3}) &#8211; (\\frac{1^2}{2} +\\frac{1^3}{3})]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= [(\\frac{4}{2} + \\frac{8}{3}) &#8211; (\\frac{1}{2} +\\frac{1}{3})]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= [\\frac{4}{2} + \\frac{8}{3} &#8211; \\frac{1}{2} -\\frac{1}{3}]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\frac{ 12 + 16 -3 &#8211; 2}{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\frac{23}{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int_1^2  ( x + x^2)\\ dx = \\frac{23}{6}}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/BvQbQmHhz98\" title=\"Definite Integrals (Excercise Problems) - Part - 1\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ \\color {red}{Evaluate\\ :\\ \\int_0^1 (3x^2\\ -\\ 2x\\ +\\ 7)\\ dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int_0^1 (3x^2\\ -\\ 2x\\ +\\ 7)\\ dx = 3\\ \\frac{x^3}{3} \\ -2\\  \\frac{x^2}{2}\\ +\\ 7x \\Biggr]_{0}^{1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= [(1^3 &#8211; 1^2 + 7(1))- 0)]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 1\\ -\\ 1 +\\ 7\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= 7\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int_0^1 (3x^2\\ -\\ 2x\\ +\\ 7)\\ dx =\\ 7}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/pR3eE5G6zbE\" title=\"Definite Integrals (Exercise Problems) - Part - 2\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ \\color {red}{Evaluate\\ :\\ \\int_1^3 (4x\\ -\\ 5x^2)\\ dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int_1^3 (4x\\ -\\ 5x^2)\\ dx = 4\\ \\frac{x^2}{2} \\ -\\ 5\\  \\frac{x^3}{3}\\  \\Biggr]_{1}^{3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= [(2(3^2)\\  -\\ 5 \\frac{3^3}{3})\\ -\\ (2(1^2)\\ -\\ 5\\ \\frac{1^3}{3})]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= [(18\\  -\\ 45)\\ -\\ (2\\ -\\ \\frac{5}{3})]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= [(-27)\\ -\\ (2\\ -\\ \\frac{5}{3})]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= [-\\ 29\\ +\\ \\frac{5}{3}]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{-\\ 82}{3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int_1^3 (4x\\ -\\ 5x^2)\\ dx =\\ \\frac{-\\ 82}{3}}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/A55Yh8X9wZQ\" title=\"Definite Integrals (Exercise) - Part - 3\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4.\\ \\color {red}{Evaluate: \\int_0^\\frac{\\pi}{4} sec^2 x\\ dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int_0^\\frac{\\pi}{4} sec^2 x\\ dx =\\ tan\\ x\\  \\Biggr]_{0}^{\\frac{\\pi}{4}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ (tan\\ \\frac{\\pi}{4} &#8211; 0)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ (1\\ -\\ 0)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int_0^\\frac{\\pi}{4} sec^2 x\\ dx   =\\ 1}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/sugGi6Ar754\" title=\"Definite Integrals (Exercise Problems) - Part - 4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"autorelaxed\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"6771622407\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\underline{PART\\ -\\ B}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5.\\ \\color {red}{Evaluate\\ :\\ \\int_1^2 (x^2\\ +\\ x\\ +\\ 1)\\ dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int_1^2 ( x^2\\ +\\ x\\ +\\ 1)\\ dx = (\\frac{x^3}{3} +\\frac{x^2}{2}\\ +\\ x) \\Biggr]_{1}^{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= [(\\frac{2^3 }{3}+\\frac{2^2}{2}\\ +\\ 2) &#8211; (\\frac{1^3}{3} +\\frac{1^2}{2}\\ +\\ 1)]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= [(\\frac{8}{3}\\ +\\ \\frac{4}{2}\\ +\\ 2) &#8211; (\\frac{1}{3}\\ +\\frac{1}{2}\\ +\\ 1)]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= [\\frac{8}{3}\\ +\\ 2\\ +\\ 2\\ -\\ \\frac{1}{3} &#8211; \\frac{1}{2} -\\ 1]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 3\\ +\\ \\frac{8}{3}\\ -\\ \\frac{1}{3} &#8211; \\frac{1}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\frac{18\\ +\\ 16-\\ 2\\ -3}{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{29}{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int_1^2  ( x^2\\ +\\ x\\ +\\ 1)\\ dx = \\frac{29}{6}}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/IFC8Ariiuls\" title=\"Definite Integrals (Exercise Problems) - Part - 5\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[6.\\ \\color {red}{Evaluate\\ :\\ \\int_0^1 \\frac{dx}{\\sqrt{1 &#8211; x^2}}}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int_0^1 \\frac{dx}{\\sqrt{1 &#8211; x^2}}\\ =\\ sin^{-1}\\ x\\ \\Biggr]_{0}^{1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=[sin^{-1}\\ (1) &#8211; sin^{-1}\\ (0)]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{\\pi}{2}\\ -\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{\\pi}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int_0^1 \\frac{dx}{\\sqrt{1 &#8211; x^2}}\\ =\\frac{\\pi}{2}}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/y21cQua7urI\" title=\"Definite Integrals (Exercise Problems) - Part - 6\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[7.\\ \\color {red}{Evaluate\\ :\\ \\int_0^\\frac{\\pi}{2} (2 + sin x)^2 cos x \\ dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\ u =2 + sin x\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx}= cos x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[du\\ =\\ cos x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int_0^\\frac{\\pi}{2} (2 + sin x)^2 cos x \\ dx = \\int_0^\\frac{\\pi}{2} u^2 \\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\Biggr[\\frac{u^3}{3}\\Biggr]_{0}^{\\frac{\\pi}{2}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\Biggr[\\frac{(2 + sin x)^3}{3}\\Biggr]_{0}^{\\frac{\\pi}{2}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=[\\frac{(2 + sin \\frac{\\pi}{2})^3}{3} &#8211; \\frac{( 2 + sin\\ 0)^3}{3}]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=[\\frac{(2 + 1)^3}{3} &#8211; \\frac{( 2 + 0)^3}{3}]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=[\\frac{(3)^3}{3} &#8211; \\frac{(2)^3}{3}]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{27}{3} &#8211; \\frac{8}{3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{27- 8}{3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{19}{3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int_0^\\frac{\\pi}{2} (2 + sin x)^2 cos x\\ dx   =\\ \\frac{19}{3}}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/3EeK2vDE2uQ\" title=\"Definite Integrals (Exercise Problems) - Part - 7\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- Ad1 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8240817448\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\underline{PART\\ -\\ C}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[8.\\ \\color{red}{Evaluate: \\int_0^\\frac{\\pi}{2} \\frac{sin^2 x}{ 1- cos x} \\ dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int_0^\\frac{\\pi}{2} \\frac{sin^2 x}{ 1- cos x} \\ dx = \\int_0^\\frac{\\pi}{2} \\frac{1 &#8211; cos^2 x}{ 1- cos x} \\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\int_0^\\frac{\\pi}{2} \\frac{(1+ cos x)(1 &#8211; cos x)}{1- cos x}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\int_0^\\frac{\\pi}{2}  ( 1 + cos x)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\Biggr[(x\\ +\\  sin x)\\Biggr]_{0}^{\\frac{\\pi}{2}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=[(\\frac{\\pi}{2}\\ +\\ sin \\frac{\\pi}{2})  &#8211; ( 0 + sin\\ 0)]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{\\pi}{2}\\ +\\ 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int_0^\\frac{\\pi}{2} \\frac{sin^2 x}{ 1- cos x} \\ dx   =\\frac{\\pi}{2}\\ +\\ 1}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/IHxl6XWK2Mg\" title=\"Definite Integrals (Exercise Problems) - Part - 8\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[9.\\ \\color{red}{Evaluate: \\int_0^\\frac{\\pi}{4} tan\\ x\\ sec^2\\ x \\ dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\ u\\ =\\ tan\\ x\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx}=\\ sec^2\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[du\\ =\\ sec^2\\ x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int_0^\\frac{\\pi}{4} tan\\ x\\ sec^2\\ x \\ dx = \\int_0^\\frac{\\pi}{4} u \\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\Biggr[\\frac{u^2}{2}\\Biggr]_{0}^{\\frac{\\pi}{4}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\Biggr[\\frac{(tan\\ x)^2}{2}\\Biggr]_{0}^{\\frac{\\pi}{4}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{(tan \\frac{\\pi}{4})^2}{2} &#8211; \\frac{( tan\\ 0)^2}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{(1)^2}{2} &#8211; \\frac{(0)^2}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{1}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int_0^\\frac{\\pi}{4} tan\\ x\\ sec^2\\ x \\ dx   =\\frac{1}{2}}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/oZ-1MICrYwM\" title=\"Definite Integrals (Exercise Problems) - Part - 9\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[10.\\ \\color{red}{Evaluate:\\ \\hspace{2cm}\\ (i)\\ \\int_0^1\\ (2x\\ +\\ 3)^4\\ dx\\ \\hspace{2cm}\\ (ii)\\ \\int_0^\\frac{\\pi}{2} cos^2 x\\ dx}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(i)\\ \\int_0^1\\ (2x\\ +\\ 3)^4\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\ u\\ =\\ 2x\\ +\\ 3\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx}\\ =\\ 2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[dx\\ = \\frac{1}{2}\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int_0^1\\ (2x\\ +\\ 3)^4\\ dx = \\frac{1}{2}\\ \\int_0^1 u^4\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{1}{2}[\\frac{u^5}{5}] \\Biggr]_{0}^{1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{1}{2}[\\frac{(2x\\ +\\ 3)^5}{5}] \\Biggr]_{0}^{1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{1}{10}[(2(1) + 3)^5\\ -\\ (2(0) + 3)^5]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{1}{10}[(5)^5\\ &#8211; (3)^5]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{1}{10}[3125\\ &#8211; 243]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{2882}{10}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 288.2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int_0^1\\ (2x\\ +\\ 3)^4 \\ dx   =\\ 288.2}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/k1ckK-XVY8Y\" title=\"Definite Integrals (Exercise Problems) - Part - 10 (i)\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\ \\int_0^\\frac{\\pi}{2} cos^2 x\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int_0^\\frac{\\pi}{2} cos^2 x\\ dx = \\int_0^\\frac{\\pi}{2} [ \\frac{1\\ +\\ cos\\ 2x}{2}]\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{1}{2}[x +\\ \\frac{sin\\ 2x}{2}] \\Biggr]_{0}^{\\frac{\\pi}{2}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{1}{2}[(\\frac{\\pi}{2}\\ +\\ 0) &#8211; ( 0\\ +\\ 0)]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{1}{2}[\\frac{\\pi}{2}]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{\\pi}{4}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int_0^\\frac{\\pi}{2} cos^2 x\\ dx   =\\frac{\\pi}{4}}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/6e5WVYvKv-Y\" title=\"Definite Integrals (Exercise Problems) - Part - 10 (ii)\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[11.\\ \\color{red}{Evaluate: \\int_0^\\frac{\\pi}{2} log(tan\\ x)\\ dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Use\\ the\\ property\\ \\int_{0}^{a} f(x)\\ dx = \\int_{0}^{a} f(a &#8211; x)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ I= \\int_0^\\frac{\\pi}{2} log(tan\\ x)\\ dx  &#8212;&#8212;&#8212;&#8212;&#8212;&#8212; (1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\int_0^\\frac{\\pi}{2} {log(tan\\ (\\frac{\\pi}{2} &#8211; x))}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\int_0^\\frac{\\pi}{2} log(cot\\ x)\\ dx&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;-( 2 )\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Adding\\ (1)\\ and\\ (2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ I= \\int_0^\\frac{\\pi}{2}  log(tan\\ x)\\ dx + \\int_0^\\frac{\\pi}{2} log(cot\\ x)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\int_0^\\frac{\\pi}{2} (log(tan\\ x)\\ +\\ log( cot\\ x))\\ dx \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\int_0^\\frac{\\pi}{2} (log(tan\\ x\\ \u00d7\\ cot\\ x))\\ dx \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\int_0^\\frac{\\pi}{2} log\\ 1\\ dx \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int_0^\\frac{\\pi}{2} log(tan\\ x)\\ dx   =\\ 0}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/P80Ph1VFsKM\" title=\"Definite Integrals (Exercise Problems) - 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