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, ∫0π 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]0π
= (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.
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FAQs
Q1: What is the integral of xcosx?
Answer: The integarl of xcosx is ∫xcosx dx = xsinx +cosx+C, where C denotes a constant.