Question

What is antiderivative of $F(x) = x\cos x$

Answer 1

Here in this question given an indefinite integral, we have to find the integrated value of given function. First rewrite the given function by using the trigonometric identity and next it can be solved by the method of integration by parts by separating the function as $u$ and $v$, later integrated by using the standard formulas of integration. And by further simplification we get the required solution.

Complete step-by-step solution:
Integration by parts is a technique for performing indefinite integration or definite integration by expanding the differential of a product of functions $d\left( {uv} \right)$ and expressing the original integral in terms of a known integral $\int {v\,du}$. A single integration by parts starts with
$d(uv) = u\,dv + v\,du$
and integrates both sides,
$\int {d(uv)} = uv = \int {u\,dv} + \int {v\,} du.$------(1)
Rearranging gives
$\int {u\,} dv = uv - \int v \,du.$---------(2)
Consider the given function $\int {x\,\cos x} \,dx$-----(3)
On applying the integration by parts that is the (2) equation
$\Rightarrow \int {x\,\cos x} \,dx = x\int {\cos x\,dx - \int {\dfrac{d}{{dx}}(x)\int {\cos x\,dx} } }$----------(4)
Here we have to know about the integration formulas
The $\int {\,\cos x} \,dx = \sin x$
And we have to know about the differentiation formulas
$\dfrac{d}{{dx}}(x) = 1$
On implementing the formula to the equation (4) we have
$\Rightarrow \int {x\,\cos x} \,dx = x\,\sin x - \int {\sin x\,dx}$------(5)
we have to know about the integration formulas
The $\int {\sin x} \,dx = - \cos x$
On implementing the formula to the equation (5) we have
$\Rightarrow \int {x\,\cos x} \,dx = x\,\sin x - ( - \cos x)$------(6)
On simplifying this we have
$\Rightarrow \int {x\,\cos x} \,dx = x\,\sin x + \cos x + c$
c is the integration constant

Note: In integration we have two kinds one is definite integral and other one is definite integral. This question comes under the indefinite integral. While integrating the function which is in the form of product or division form we use the integration by parts method. By applying the integration by parts we obtain the solution.