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MathematicsU-Substitution
Teaching

Problem

∫2x(x2+1)3 dx\int 2x(x^2 + 1)^3 \, dx
1
Let u=x2+1\text{Let } u = x^2 + 1

Choose the inner function

2
du=2x dxdu = 2x \, dx

Differentiate both sides

3
∫u3 du\int u^3 \, du

Substitute — much simpler!

4
=u44+C=(x2+1)44+C= \frac{u^4}{4} + C = \frac{(x^2+1)^4}{4} + C

Substitute back

Chat
Live
TutorWe substitute u=x2+1u = x^2+1 because its derivative 2x2x already appears in the integrand.
Why did we pick u=x2+1u = x^2+1 and not something else?
TutorLook at the integrand: 2x2x is the derivative of x2+1x^2+1. When you see a function and its derivative together — that's your signal.
Now try: ∫3x2cos⁡(x3) dx\int 3x^2 \cos(x^3) \, dx
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