∫xcosx+12x3esinx+x2dx

$\int \frac{x\mathrm{cos}x+1}{\sqrt{2{x}^{3}{e}^{\mathrm{sin}x}+{x}^{2}}}dx$

1. A

$\mathrm{log}\left|{\mathrm{tan}}^{-1}\left(x{e}^{\mathrm{sin}x}\right)\right|+c$

2. B

$\mathrm{log}\left|\frac{\sqrt{2x{e}^{\mathrm{sin}x}+1}-1}{\sqrt{2x{e}^{\mathrm{sin}x}+1}+1}\right|+c$

3. C

$\mathrm{log}\left(\sqrt{2x{e}^{\mathrm{sin}x}}\right)+c$

4. D

none of these

Register to Get Free Mock Test and Study Material

+91

Verify OTP Code (required)

Solution:

$I=\int \frac{x\mathrm{cos}x+1}{\sqrt{2{x}^{3}{e}^{\mathrm{sin}x}+{x}^{2}}}dx$

$=\int \frac{x\mathrm{cos}x+1}{x\sqrt{2x{e}^{\mathrm{sin}x}+1}}\text{\hspace{0.17em}}dx$

Put $2x{e}^{\mathrm{sin}x}+1=t$

$⇒\left(2x\left({e}^{\mathrm{sin}x}\right)\mathrm{cos}x+{e}^{\mathrm{sin}x}\left(2\right)\right)dx=dt$

$⇒2{e}^{\mathrm{sin}x}\left[x\mathrm{cos}x+1\right]dx=dt$

$⇒2x{e}^{\mathrm{sin}x}\left(\frac{x\mathrm{cos}x+1}{x}\right)dx=dt$

$⇒\left(t-1\right)\left(\frac{x\mathrm{cos}x+1}{x}\right)dx=dt$

$⇒\left(\frac{x\mathrm{cos}x+1}{x}\right)dx=\frac{dt}{\left(t-1\right)}$

$t={z}^{2}⇒dt=2z\text{\hspace{0.17em}}dz$

$I=\int \frac{2z}{\left({z}^{2}-1\right)\left(z\right)}dz$

$=2\left(\frac{1}{2}\right)\mathrm{ln}\left|\frac{z-1}{z+1}\right|+c$

$I=\mathrm{ln}\left|\frac{\sqrt{2x{e}^{\mathrm{sin}x}+1}-1}{\sqrt{2x{e}^{\mathrm{sin}x}+1}+1}\right|+c$

Register to Get Free Mock Test and Study Material

+91

Verify OTP Code (required)