NIMCET 2017 — Mathematics PYQ

Concept:
Integration by parts: Integration by parts is a method to find integrals of products.  

• The formula for integrating by parts is given by,  
$\int uvdx=u\int vdx-\int u^{\prime }\left(\int vdx\right)dx$  

Where $u$ is the function $u(x)$ and $v$ is the function $v(x)$.  

ILATE Rule: Usually, the preference order of this rule is based on some functions such as Inverse, Logarithmic, Algebraic, Trigonometric, and Exponential.

Calculation:

Let $I={\int }_0^{\pi }{x}^3\sin xdx$

Apply by parts rule, we get  

$=x^3{\int }_0^{\pi }\sin xdx-{\int }_0^{\pi }3x^2(-\cos x)dx$

$=[x^3(-\cos x){]}_0^{\pi }+3{\int }_0^{\pi }{x}^2\cos xdx-{\int }_0^{\pi }2x(\sin x)dx$

$={\pi }^3+0-6{\int }_0^{\pi }x(\sin x)dx$

$={\pi }^3-6[x{\int }_0^{\pi }\sin xdx-{\int }_0^{\pi }(-\cos x)dx]$

$={\pi }^3-6[\pi -0]$

$={\pi }^3-6\pi$

Hence, option (1) is correct.