{"id":29448,"date":"2022-05-02T12:44:21","date_gmt":"2022-05-02T07:14:21","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=29448"},"modified":"2024-05-13T17:20:14","modified_gmt":"2024-05-13T11:50:14","slug":"bernoullis-formulaexcercise-problems-with-solutions","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=29448","title":{"rendered":"BERNOULLI\u2019S FORMULA(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 x^2\\ sin\\ 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\">\\[ ILATE \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= x^2\\ \\hspace{2cm}\\  dv = sin\\ x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[v = &#8211; cos x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u^! = 2x\\ \\hspace{2cm}\\ v_1 = &#8211; sin\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u^{!!} = 2\\ \\hspace{2cm}\\  v_2 = cos \\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; u^!v_1 + u^{!!}v_2 \\ +\\ c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x^2\\ sin\\ x\\ dx = x^2(-cos x)\\ -\\ 2x (- sin\\ x)\\ +\\ 2 (cos\\ x)\\  + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int x^2\\ sin\\ x\\ dx = x^2(-cos x)\\ -\\ 2x (- sin\\ x)\\ +\\ 2 (cos\\ x)+c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/W4DIWFfq_7c\" title=\"Bernoulli's Formula (Exercise 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 x^2\\ e^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\">\\[  ILATE  \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= x^2\\ \\hspace{2cm}\\  dv = e^x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u^! = 2x\\ \\hspace{2cm}\\  v\\ =\\ e^x  \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u^{!!} = 2\\ \\hspace{2cm}\\  v_1 =  e^x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\  v_2 =  e^x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; u^!v_1 + u^{!!}v_2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x^2\\ e^x\\ dx = x^2\\ e^x\\ &#8211; 2x\\ e^x\\ +\\ 2\\ e^x\\  + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int x^2\\ e^x\\ dx = x^2\\ e^x\\ &#8211; 2x\\ e^x\\ +\\ 2\\ e^x+c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/-sOGL4EsFms\" title=\"Bernoulli's Formula (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<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\\ -\\ B}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ \\color {red}{Evaluate\\ :\\ \\int x^2\\ cos\\ 3x\\ 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\">\\[  ILATE  \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= x^2\\ \\hspace{2cm}\\  dv = cos\\ 3x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u^! = 2x\\ \\hspace{2cm}\\  v =  \\frac{sin\\ 3x}{3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u^{!!} = 2\\ \\hspace{2cm}\\  v_1 = &#8211; \\frac{cos\\ 3x}{9}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\  v_2 =  -\\frac{sin\\ 3x}{27}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; u^!v_1 + u^{!!}v_2 \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x^2\\ cos\\ 3x\\ dx = x^2(\\frac{sin\\ 3x}{3}) &#8211; 2x (-\\frac{cos\\ 3x}{9}) + 2(- \\frac{sin\\ 3x}{27}) + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int x^2\\ cos\\ 3x\\ dx = x^2\\frac{sin\\ 3x}{3}\\ +\\ 2x \\frac{cos\\ 3x}{9}\\ -\\ 2\\frac{sin\\ 3x}{27}+c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/UVozef1Kbow\" title=\"Bernoulli's Formula (Exercise Problems) - Part - 3\" 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\\ -\\ C}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4.\\ \\color{red}{Evaluate:\\ \\hspace{2cm}\\ (i)\\ \\int\\ x^2\\ sin\\ 4x\\ dx\\ \\hspace{2cm}\\ (ii)\\ \\int x^2\\ e^{-7x}\\ 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\\ x^2\\ sin\\ 4x\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  ILATE  \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= x^2\\ \\hspace{2cm}\\  dv = sin\\ 4x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u^! = 2x\\ \\hspace{2cm}\\  v = &#8211; \\frac{cos\\ 4x}{4}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u^{!!} = 2\\ \\hspace{2cm}\\  v_1 = &#8211; \\frac{sin\\ 4x}{16}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\  v_2 =  \\frac{cos\\ 4x}{64}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; u^!v_1 + u^{!!}v_2 \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x^2\\ sin\\ 4x\\ dx = x^2(-\\frac{cos\\ 4x}{4}) &#8211; 2x (-\\frac{sin\\ 4x}{16}) + 2(\\frac{cos\\ 4x}{16}) + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = &#8211; x^2\\frac{cos\\ 4x}{4} + x \\frac{sin\\ 4x}{8} + \\frac{cos\\ 4x}{8} + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int x^2\\ sin\\ 4x\\ dx = &#8211; x^2\\frac{cos\\ 4x}{4} + x \\frac{sin\\ 4x}{8} + \\frac{cos\\ 4x}{8}+c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/16uJcdTEcW8\" title=\"Bernoulli's Formula (Exercise Problems) - Part - 4 (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\\ x^2\\ e^{-7x}\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  ILATE  \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= x^2\\ \\hspace{2cm}\\  dv = e^{-7x}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u^! = 2x\\ \\hspace{2cm}\\  v =  \\frac{e^{-7x}}{-7}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u^{!!} = 2\\ \\hspace{2cm}\\  v_1 =  \\frac{e^{-7x}}{49}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\  v_2 =  \\frac{e^{-7x}}{-343}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; u^!v_1 + u^{!!}v_2 \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x^2\\ e^{-7x}\\ dx = x^2\\frac{e^{-7x}}{-7}\\ -\\ 2\\ x \\frac{e^{-7x}}{49}\\ +\\ 2\\frac{e^{-7x}}{-343}\\ +\\ c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = -\\ x^2\\frac{e^{-7x}}{7}\\ -\\ 2\\ x \\frac{e^{-7x}}{49}\\ -\\ 2\\frac{e^{-7x}}{343} + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int x^2\\ e^{-7x}\\ dx = -\\ x^2\\frac{e^{-7x}}{7}\\ -\\ 2\\ x \\frac{e^{-7x}}{49}\\ -\\ 2\\frac{e^{-7x}}{343}+c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/6_A5xTQ1x9A\" title=\"Bernoulli's Formula (Exercise Problems) - Part - 4 (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\">\\[5.\\ \\color{red}{Evaluate:\\ \\hspace{2cm}\\ (i)\\ \\int\\ x^2\\ cos\\ 5x\\ dx\\ \\hspace{2cm}\\ (ii)\\ \\int x^2\\ e^{-2x}\\ 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\\ x^2\\ cos\\ 5x\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  ILATE  \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= x^2\\ \\hspace{2cm}\\  dv = cos\\ 5x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u^! = 2x\\ \\hspace{2cm}\\  v =  \\frac{sin\\ 5x}{5}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u^{!!} = 2\\ \\hspace{2cm}\\  v_1 = &#8211; \\frac{cos\\ 5x}{25}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\  v_2 =  -\\frac{sin\\ 5x}{125}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; u^!v_1 + u^{!!}v_2 \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x^2\\ cos\\ 5x\\ dx = x^2(\\frac{sin\\ 5x}{5}) &#8211; 2x (-\\frac{cos\\ 5x}{25}) + 2(- \\frac{sin\\ 5x}{125}) + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =  x^2\\frac{sin\\ 2x}{2}\\ +\\ 2x \\frac{cos\\ 5x}{25} &#8211; 2\\frac{sin\\ 5x}{125} + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int x^2\\ cos\\ 5x\\ dx = x^2\\frac{sin\\ 2x}{2}\\ +\\ 2x \\frac{cos\\ 5x}{25} &#8211; 2\\frac{sin\\ 5x}{125}+c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/-wn1Xw1SYRk\" title=\"Bernoulli&#39;s Formula (Exercise Problems) - Part - 5 (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\\ x^2\\ e^{-2x}\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  ILATE  \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= x^2\\ \\hspace{2cm}\\  dv = e^{-2x}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u^! = 2x\\ \\hspace{2cm}\\  v =  \\frac{e^{-2x}}{-2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u^{!!} = 2\\ \\hspace{2cm}\\  v_1 =  \\frac{e^{-2x}}{4}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\  v_2 =  \\frac{e^{-2x}}{-8}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; u^!v_1 + u^{!!}v_2 \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x^2\\ e^{-2x}\\ dx = x^2\\frac{e^{-2x}}{-2}\\ -\\ 2\\ x \\frac{e^{-2x}}{4}\\ +\\ 2\\frac{e^{-2x}}{-8}\\ +\\ c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = -\\ x^2\\frac{e^{-2x}}{2}\\ -\\  x \\frac{e^{-2x}}{2}\\ -\\ \\frac{e^{-2x}}{4} + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int x^2\\ e^{-2x}\\ dx = -\\ x^2\\frac{e^{-2x}}{2}\\ -\\  x \\frac{e^{-2x}}{2}\\ -\\ \\frac{e^{-2x}}{4}+c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/mTnZozdanr0\" title=\"Bernoulli&#39;s Formula (Exercise Problems) - Part - 5 (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\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[7.\\ \\color{red}{Evaluate:\\ \\hspace{2cm}\\ (i)\\ \\int\\ x^2\\ e^{-3x}\\ dx\\ \\hspace{2cm}\\ (ii)\\ \\int x^2\\ sin\\ 6x\\ 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\\ x^2\\ e^{-3x}\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  ILATE  \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= x^2\\ \\hspace{2cm}\\  dv = e^{-3x}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u^! = 2x\\ \\hspace{2cm}\\  v =  \\frac{e^{-3x}}{-3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u^{!!} = 2\\ \\hspace{2cm}\\  v_1 =  \\frac{e^{-3x}}{9}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\  v_2 =  \\frac{e^{-3x}}{-27}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; u^!v_1 + u^{!!}v_2 \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x^2\\ e^{-3x}\\ dx = x^2\\frac{e^{-3x}}{-3}\\ -\\ 2\\ x \\frac{e^{-3x}}{9}\\ +\\ 2\\frac{e^{-3x}}{-27}\\ +\\ c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = -\\ x^2\\frac{e^{-3x}}{3}\\ -\\  2x \\frac{e^{-3x}}{9}\\ -\\ 2\\frac{e^{-3x}}{27} + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int x^2\\ e^{-3x}\\ dx = -\\ x^2\\frac{e^{-3x}}{3}\\ -\\  2x \\frac{e^{-3x}}{9}\\ -\\ 2\\frac{e^{-3x}}{27}+c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/mTnZozdanr0\" title=\"Bernoulli&#39;s Formula (Exercise Problems) - Part - 7 (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\\ x^2\\ sin\\ 6x\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  ILATE  \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= x^2\\ \\hspace{2cm}\\  dv = sin\\ 6x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u^! = 2x\\ \\hspace{2cm}\\  v = &#8211; \\frac{cos\\ 6x}{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u^{!!} = 2\\ \\hspace{2cm}\\  v_1 = &#8211; \\frac{sin\\ 6x}{36}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\  v_2 =  \\frac{cos\\ 6x}{216}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; u^!v_1 + u^{!!}v_2 \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x^2\\ sin\\ 6x\\ dx = x^2(-\\frac{cos\\ 6x}{6}) &#8211; 2x (-\\frac{sin\\ 6x}{36}) + 2(\\frac{cos\\ 6x}{216}) + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = &#8211; x^2\\frac{cos\\ 6x}{6} + x \\frac{sin\\ 6x}{18} + \\frac{cos\\ 6x}{108} + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int x^2\\ sin\\ 6x\\ dx = &#8211; x^2\\frac{cos\\ 6x}{6} + x \\frac{sin\\ 6x}{18} + \\frac{cos\\ 6x}{108}+c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/g4s7Wp10ohc\" title=\"Bernoulli&#39;s Formula (Exercise Problems ) - 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