Calculus / Unit 8: The Fundamental Theorem of Calculus
8.4 u-Substitution (Indefinite Integrals)
Notes
u-substitution reverses the chain rule. Let $u$ be an inside function whose derivative also appears.
- Steps: choose $u$, compute $du = u'\,dx$, rewrite the integral in terms of $u$, integrate, then replace $u$.
- Example: $\displaystyle \int 2x(x^2 + 1)^3\,dx$. Let $u = x^2 + 1$, $du = 2x\,dx$. Then $\displaystyle \int u^3\,du = \frac{u^4}{4} + C = \frac{(x^2 + 1)^4}{4} + C$.
- If $du$ is off by a constant, adjust: $\displaystyle \int \cos(3x)\,dx = \frac{1}{3}\sin(3x) + C$.
- Check by differentiating.
Practice
Find $\displaystyle \int 2x(x^2 + 1)^3\,dx$.
With $u = x^2 + 1$, the integral is $\displaystyle \int u^3\,du$.
Find $\displaystyle \int \cos(3x)\,dx$.
With $u = 3x$, $\displaystyle dx = \frac{du}{3}$.
Find $\displaystyle \int x e^{x^2}\,dx$.
With $u = x^2$, $\displaystyle x\,dx = \frac{du}{2}$, giving $\displaystyle \frac{1}{2}e^u$.
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