Step by step
The question
Question: find the integral of sin⁴ x cos x.
∫ sin4 x cos x dx =
1 Choose u
Let u = sin x.
Its derivative is cos x: du = cos x dx.
And cos x dx is right there in the integral.
u = sin x, du = cos x dx
2 Write it with u
sin x becomes u, and cos x dx becomes du.
= ∫ u4 du
Careful!
cos x dx becomes du.
It does not stay in the integral as well.
Careful: cos x dx → du: it is used up
3 Integrate
The power rule: add 1 to each power, divide by the new power.
= u55 + C
4 Put x back
Put sin x back in place of u.
So the answer is sin⁵ x/5 + C.
= sin5 x5 + C
5 Check
Let's check: the slope of our answer F near x = 1, with a calculator in radians: about 0.271.
F(1.001) − F(0.999)0.002 ≈ 0.271
The question at x = 1 gives 0.271.
The same: the slope of the answer is the question.
Right!
sin⁴ 1 cos 1 ≈ 0.271
Let's look back.
First: choose u.
Then: write it with u.
Then: integrate.
Then: put x back.
Last: check.
Answer
= sin5 x5 + C