The integral of xcosx is equal to xsinx +cosx+C where C is an arbitrary constant, and it is denoted by ∫xcosx dx. The function xcosx is a product of two functions x and cosx. So we can use integration by parts formula to find its integration.

Notation of Integral of xcosx: | ∫xcosx dx |

Integration formula of xcosx: | ∫xcosx dx = xsinx +cosx+C. |

Definite Integral of xcosx: | The definite integral of xcosx from 0 to π is equal to -2, that is, ∫_{0}^{π} xcosx dx = -2. |

Let us now find the integration of xcosx dx.

## What is the Integration of xcosx?

**Answer:** The integration of xcosx is xsinx +cosx+C, C is a constant.

**Explanation:**

As xcosx is a product of two functions x and cosx. So the integration of xcosx is computed using the integration by parts: This rule says that if u and v are two functions of x, then the integral of uv is given by the formula below:

∫uv dx = u ∫v dx – ∫[$\frac{du}{dx}$∫v dx] dx. |

Here put u=x and v=cosx.

So the integration of xcosx is

∫xcosx dx

= x ∫cosx dx – ∫ $\big[\dfrac{d}{dx}(x) \int \cos x\ dx \big]dx$

= x sinx – ∫ sinx dx +C as ∫cosx dx = sinx

= x sinx + cosx +C as ∫sinx dx = -cosx.

So the integration of xcosx is equal to x sinx + cosx +C where C is an integral constant, and this is obtained by using the integration by parts formula.

## Definite Integral of xcosx From 0 to π

Question: Find the definite integral of xcosx from 0 to π, that is find$\int_0^\pi$ xcosx dx. |

**Answer:**

From above, $\int_0^\pi$ xcosx dx

= [x sinx + cosx]_{0}^{π}

= (π sinπ + cosπ) – (0+cos0)

= -1 -1 as we know that cos0=1, cosπ=-1 and sinπ =0.

= -2.

So the integral of xcosx from 0 to π is equal to -2.

**Read Also:**

## FAQs

### Q1: What is the integral of xcosx?

Answer: The integarl of xcosx is ∫xcosx dx = xsinx +cosx+C, where C denotes a constant.