Question

`int sin^(5)x cos^(1/2)xdx`
please give me its answer only​

Answer 1

♣ Qᴜᴇꜱᴛɪᴏɴ :

$\large\boxed{\sf{\int \sin ^5\left(x\right)\cos ^{\tfrac{1}{2}}\left(x\right)dx}}$

♣ ᴀɴꜱᴡᴇʀ :

$\boxed{\sf{\int \sin ^5\left(x\right)\cos ^{\tfrac{1}{2}}\left(x\right)dx=-\frac{2}{3}\cos ^{\tfrac{3}{2}}\left(x\right)+\frac{4}{7}\cos ^{\tfrac{7}{2}}\left(x\right)-\frac{2}{11}\cos ^{\tfrac{11}{2}}\left(x\right)+C}}$

♣ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴꜱ :

$\int \sin ^5\left(x\right)\sqrt{\cos \left(x\right)}dx$

$\sin ^5\left(x\right)=\sin ^4\left(x\right)\sin \left(x\right)$

$=\int \sin ^4\left(x\right)\sin \left(x\right)\sqrt{\cos \left(x\right)}dx$

$\sin ^4\left(x\right)=\left(\sin ^2\left(x\right)\right)^2$

$=\int \left(\sin ^2\left(x\right)\right)^2\sin \left(x\right)\sqrt{\cos \left(x\right)}dx$

$\mathrm{Use\:the\:following\:identity}:\quad \sin ^2\left(x\right)=1-\cos ^2\left(x\right)$

$=\int \left(1-\cos ^2\left(x\right)\right)^2\sin \left(x\right)\sqrt{\cos \left(x\right)}dx$

$\text { Apply u - substitution: } u=\cos (x)$

$=\int \:-\sqrt{u}\left(1-u^2\right)^2du$

$\text { Expand }-\sqrt{u}\left(1-u^{2}\right)^{2}:-\sqrt{u}+2 u^{\tfrac{5}{2}}-u^{\tfrac{9}{2}}$

$=\int \:-\sqrt{u}+2u^{\tfrac{5}{2}}-u^{\tfrac{9}{2}}du$

$\mathrm{Apply\:the\:Sum\:Rule}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx$

$=-\int \sqrt{u}du+\int \:2u^{\tfrac{5}{2}}du-\int \:u^{\tfrac{9}{2}}du$

$\begin{array}{l}\int \sqrt{u} d u=\dfrac{2}{3} u^{\tfrac{3}{2}} \\\\\int 2 u^{\tfrac{5}{2}} d u=\dfrac{4}{7} u^{\tfrac{7}{2}} \\\\\int u^{\tfrac{9}{2}} d u=\dfrac{2}{11} u^{t\frac{11}{2}}\end{array}$

$=-d\frac{2}{3}u^{\tfrac{3}{2}}+\dfrac{4}{7}u^{\tfrac{7}{2}}-\dfrac{2}{11}u^{\tfrac{11}{2}}$

$\mathrm{Substitute\:back}\:u=\cos \left(x\right)$

$=-\dfrac{2}{3}\cos ^{\tfrac{3}{2}}\left(x\right)+\dfrac{4}{7}\cos ^{\tfrac{7}{2}}\left(x\right)-\dfrac{2}{11}\cos ^{\tfrac{11}{2}}\left(x\right)$

$\mathrm{Add\:a\:constant\:to\:the\:solution}$

$\boxed{\sf{=-\frac{2}{3}\cos ^{\tfrac{3}{2}}\left(x\right)+\frac{4}{7}\cos ^{\tfrac{7}{2}}\left(x\right)-\frac{2}{11}\cos ^{\tfrac{11}{2}}\left(x\right)+C}}$