Integrate ∫ x sin x dx Step by Step

Step by step

The question

Question: find the integral of x sin x.

∫ x sin x dx =

1 Choose u and dv

Choose u = x: its derivative is simpler, just a number.

dv is the rest: sin x dx, which we can integrate.

u = x, dv = sin x dx

2 Find du and v

du: the derivative of u, times dx.

v: the integral of dv.

sin gives minus cos.

du = 1 dx, v = −cos x

3 Use the rule

The rule: the integral of u dv is u v, minus the integral of v du.

It comes from the product rule, backwards.

∫ u dv = uv − ∫ v du

u times v, minus the integral of v times du.

= −x cos x − ∫ (−cos x) dx

4 The new integral

The new integral is easy: just a number times cos.

cos integrates to sin.

= −x cos x − (−sin x)

Careful!

It is u v minus the integral of v du.

Not plus!

Careful: not uv + ∫ v du ✗

Open the bracket: a minus in front of a bracket changes the sign of every part inside.

Add C.

So the answer is −x cos x + sin x + C.

= −x cos x + sin x + C

5 Check

Let's check: take the derivative of our answer.

The product rule on each term: the first's slope times the second, plus the first times the second's slope.

derivative: −cos x + x sin x + cos x

The other terms cancel, and we get x sin x: the question again.

Right!

= x sin x

Let's look back.

First: choose u and dv.

Then: find du and v.

Then: use the rule.

Then: the new integral.

Last: check.

Answer