Q. 12 - Calculus

Question

Show that $F\left(x\right)=\sin x-x\cos x+2$ is an antiderivative of $f\left(x\right)=\sin x$. 

Step-by-Step Solution

Answer

It is shown that $F\left(x\right)=\sin x-x\cos x+2\text{ is an antiderivative of }f\left(x\right)=\sin x.$

Step 1. Given Information.

The given derivative is $F\left(x\right)=\sin x-x\cos x+2$ and the given function is $f\left(x\right)=\sin x$.

Step 2. Differentiation.

On differentiating the function,

We get,

$$\begin{gathered} \frac{d}{dx}x\sin x \\ =1\left(\sin x\right)-x\left(\cos x\right) \\ =\sin x-x\cos x \\ So, \\ \int x\sin xdx=\sin x-x\cos x+C \\ Therefore, \\ F\left(x\right)=\sin x-x\cos x+2 is an anti - derivative of f\left(x\right)=x\sin x. \end{gathered}$$