Integral of xsin(x)
The integral of xsin(x) is sin(x)−xcos(x)+C.
Graph of f(x) = x * sin(x) with the area from 0 to π shaded. The shaded area equals the definite integral, about 3.1415927.How to work it out
- 1
We want an antiderivative of f(x) = x * sin(x).
∫xsin(x)dx - 2
Method: integration by parts with u = x, dv = sin x dx.
- 3
That gives the antiderivative; add the constant C because any constant disappears when you differentiate:
∫xsin(x)dx=sin(x)−xcos(x)+C - 4
Check by differentiating the answer; you must get the original function back:
dxd[sin(x)−xcos(x)]=cos(x)−(cos(x)+x−sin(x)) - 5
As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and π:
∫0πxsin(x)dx=[sin(x)−xcos(x)]0π≈3.1415927
LaTeX source of the formulas on this page
\[ \int x \sin\left(x\right)\,dx = \sin\left(x\right) - x \cos\left(x\right) + C \]
\[ \int x \sin\left(x\right)\,dx \]
\[ \frac{d}{dx}\left[\sin\left(x\right) - x \cos\left(x\right)\right] = \cos\left(x\right) - \left(\cos\left(x\right) + x -\sin\left(x\right)\right) \]
\[ \int_{0}^{\pi} x \sin\left(x\right)\,dx = \Big[\sin\left(x\right) - x \cos\left(x\right)\Big]_{0}^{\pi} \approx 3.1415927 \]Definite integrals of the same function
| Interval | Value |
|---|
| from 0 to π | 3.1415927 |
| from 0 to π/2 | 1 |
Questions
What is the integral of x * sin(x)?
The integral of xsin(x) is sin(x)−xcos(x)+C.
Why do we add + C?
Every function F(x) + C has the same derivative, so the antiderivative is only defined up to a constant.
What is the integral of x * sin(x) from 0 to π?
About 3.1415927.