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Products and integration by parts

Integral of x sin x: Formula, Steps and Examples

The integral of x sin x is sin x − x cos x + C. See the integration by parts steps, a check by differentiating, a graph and examples like x² sin x.

Products and integration by parts Updated 3 min read

At a glance

\int x\sin x\,dx = \sin x - x\cos x + C
Antiderivative
\sin x - x\cos x + C
Method
Integration by parts with u = x, dv = \sin x\,dx
From 0 to π
\pi
From 0 to π/2
1
With sin(kx)
\int x\sin kx\,dx = -\frac{x}{k}\cos kx + \frac{1}{k^2}\sin kx + C
On this page
  1. Graph of x sin x
  2. How to integrate x sin x step by step
  3. Check by differentiating
  4. Definite integrals
  5. Worked examples
  6. Common mistakes

The integral of x\sin x is \sin x - x\cos x + C. It is a power of x times a trig function, which is a classic integration by parts problem.

\int x\sin x\,dx = \sin x - x\cos x + C

Graph of x sin x

y = x sin x with the area from 0 to π shaded
The shaded hump has area exactly \pi. The next hump, from \pi to 2\pi, lies below the axis with signed area -3\pi, and each later hump grows by 2\pi.

How to integrate x sin x step by step

The integration by parts formula is \int u\,dv = uv - \int v\,du.

  1. Choose the parts. u = x and dv = \sin x\,dx. Algebraic comes before trig in LIATE.
  2. Find du and v. du = dx and v = -\cos x, from the integral of sin x.
  3. Apply the formula. \int x\sin x\,dx = x(-\cos x) - \int(-\cos x)\,dx = -x\cos x + \int\cos x\,dx
  4. Finish. \int\cos x\,dx = \sin x, so \int x\sin x\,dx = -x\cos x + \sin x + C

If you choose u = \sin x and dv = x\,dx instead, the leftover integral is \int\frac{x^2}{2}\cos x\,dx, with a higher power. That is the signal to switch.

Check by differentiating

\frac{d}{dx}\left(\sin x - x\cos x\right) = \cos x - \left(\cos x - x\sin x\right) = x\sin x

The \cos x terms cancel, and the product rule on x\cos x supplies the x\sin x.

Definite integrals

Let F(x) = \sin x - x\cos x.

Integral Work Value
\int_0^{\pi/2} x\sin x\,dx F\!\left(\frac{\pi}{2}\right) - F(0) = 1 - 0 1
\int_0^{\pi} x\sin x\,dx F(\pi) - F(0) = \pi - 0 \pi
\int_\pi^{2\pi} x\sin x\,dx F(2\pi) - F(\pi) = -2\pi - \pi -3\pi
\int_0^{2\pi} x\sin x\,dx F(2\pi) - F(0) -2\pi

In general the hump from k\pi to (k+1)\pi has area (2k+1)\pi, with sign (-1)^k. The total area (not signed) from 0 to 2\pi is \pi + 3\pi = 4\pi.

Worked examples

Example 1: a scaled angle. \int x\sin 2x\,dx. With u = x, dv = \sin 2x\,dx, v = -\frac{1}{2}\cos 2x:

-\frac{x}{2}\cos 2x + \frac{1}{2}\int\cos 2x\,dx = -\frac{x}{2}\cos 2x + \frac{1}{4}\sin 2x + C

Example 2: x squared, with the tabular method. For \int x^2\sin x\,dx, differentiate x^2 down to 0 and integrate \sin x alongside, with alternating signs:

Sign Differentiate Integrate
+ x^2 \sin x
- 2x -\cos x
+ 2 -\sin x
0 \cos x

Multiply along the diagonals:

\int x^2\sin x\,dx = -x^2\cos x + 2x\sin x + 2\cos x + C

Example 3: a volume. Rotating the first hump of y = \sin x (from 0 to \pi) about the y-axis, the shell method gives

V = 2\pi\int_0^\pi x\sin x\,dx = 2\pi\cdot\pi = 2\pi^2

See the volumes of revolution guide for the setup.

Example 4: a shifted angle. \int x\sin(x + 1)\,dx = \sin(x+1) - x\cos(x+1) + C, since the inside has slope 1.

Common mistakes

  1. Getting v wrong. \int\sin x\,dx = -\cos x, not \cos x. Most errors on this problem are sign errors.
  2. Losing the double negative. -\int(-\cos x)\,dx = +\int\cos x\,dx.
  3. Choosing u = \sin x. The leftover integral then has x^2 in it.
  4. Forgetting \frac{1}{k} and \frac{1}{k^2} with \sin kx. Differentiate your answer to check.

Compare with the integral of x cos x, which has the same steps with different signs. More practice is in the integration by parts guide.

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Frequently asked questions

What is the integral of x sin x?
It is \sin x - x\cos x + C.

What is the integral of x sin x from 0 to π?
It is \pi. The antiderivative is \sin\pi - \pi\cos\pi = \pi at the top and 0 at the bottom.

What is the integral of x² sin x?
It is -x^2\cos x + 2x\sin x + 2\cos x + C. Use parts twice, or the tabular method.

What is the integral of x sin(2x)?
It is -\frac{x}{2}\cos 2x + \frac{1}{4}\sin 2x + C. Use parts with u = x and v = -\frac{1}{2}\cos 2x.

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Related integrals

Integral of x cos x \int x\cos x\,dx = x\sin x + \cos x + C The integral of x cos x is x sin x + cos x + C. See the integration by parts steps, a check, a graph, definite integrals and examples like x² cos x. Integral of x eˣ \int x e^x\,dx = (x - 1)e^x + C The integral of x eˣ is (x − 1)eˣ + C. See the integration by parts steps, why u = x is the right choice, a graph and examples like x e^(2x). Integral of sin x \int \sin x\,dx = -\cos x + C The integral of sin x is -cos x + C. See where the minus sign comes from, the area of one arch, sin(ax + b) by substitution and worked definite integrals.