Step by step
The question
Question: find the integral of sin x cos⁴ x.
∫ sin x cos4 x dx =
1 Choose u
Let u = cos x.
Its derivative is minus sin x: du = −sin x dx.
So sin x dx is −du.
u = cos x, du = −sin x dx
2 Write it with u
cos x becomes u, and sin x dx becomes −du.
= ∫ −u4 du
Careful!
The derivative of cos x is minus sin x.
So sin x dx is minus du: keep the minus.
Careful: not sin x dx = du ✗
3 Integrate
The power rule: add 1 to each power, divide by the new power.
= −u55 + C
4 Put x back
Put cos x back in place of u.
So the answer is −cos⁵ x/5 + C.
= −cos5 x5 + C
5 Check
Let's check: the slope of our answer F near x = 1, with a calculator in radians: about 0.072.
F(1.001) − F(0.999)0.002 ≈ 0.072
The question at x = 1 gives 0.072.
The same: the slope of the answer is the question.
Right!
sin 1 cos⁴ 1 ≈ 0.072
Let's look back.
First: choose u.
Then: write it with u.
Then: integrate.
Then: put x back.
Last: check.
Answer
= −cos5 x5 + C