Question

The integral of the function $x\sin x$ is

• A. $x\cos x-\sin x$
• B. $x\cos x+\sin x$
• C. $-x\cos x-\sin x$
• D. $-x\cos x+\sin x$

Answer 1

The correct option is D $-x\cos x+\sin x$

The given function is a product of two functions. For functions u and v,
$\int uvdx=u\int vdx-\int (\frac{du}{dx}\int vdx)dx$
Here, according to ILATE, we should take the algebraic function, x, as the first function and integrate
$\Rightarrow \int x\sin xdx=x\int \sin xdx-\int \left[\frac{d}{dx}\right(x)\int \sin xdx]dx$
=$x(-\cos x)-\int \left[\right(1\left)\right(-\cos x\left)\right]dx$
=$-x\cos x+\int \cos xdx$
=$-x\cos x+\sin x$