Integral of xcosx | Find ∫xcosx dx

The integral of xcosx is equal to xsinx +cosx+C where C is an arbitrary constant, and it is denoted by ∫xcosx dx. So the integration formula of xcosx is given by

∫xcosx dx = xsinx +cosx+C.

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.

Question-Answer

Question: Find the definite integral $\int_0^\pi$ 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:

Integration of 1/2x

Integration of ln(x)

Integration of 0

Integration of 1

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