Integrate ∫ x sin(2x) dx Step by Step

Step by step

The question

Question: find the integral of x sin(2x).

∫ x sin(2x) dx =

1 Choose u and dv

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

dv is the rest: sin(2x) dx, which we can integrate.

u = x, dv = sin(2x) dx

2 Find du and v

du: the derivative of u, times dx.

v: the integral of dv.

sin gives minus cos, over 2.

du = 1 dx, v = −12 cos(2x)

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.

= −12x cos(2x) − ∫ (−12 cos(2x)) dx

4 The new integral

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

cos integrates to sin.

= −12x cos(2x) − (−14 sin(2x))

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 −1/2 × x cos(2x) + 1/4 × sin(2x) + C.

= −12x cos(2x) + 14 sin(2x) + 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: −12 cos(2x) + x sin(2x) + 12 cos(2x)

The other terms cancel, and we get x sin(2x): the question again.

Right!

= x sin(2x)

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