Calculus / Unit 8: The Fundamental Theorem of Calculus
8.6 Choosing u and Checking Your Work
Notes
Pick $u$ as the "inside" piece (under a power, root, exponent, or in a denominator) whose derivative is also present, up to a constant.
- $\displaystyle \int x^2 e^{x^3}\,dx$: let $u = x^3$, so $\displaystyle x^2\,dx = \frac{du}{3}$. The answer is $\displaystyle \frac{1}{3}e^{x^3} + C$.
- $\displaystyle \int \frac{3x^2}{x^3 + 5}\,dx = \ln|x^3 + 5| + C$.
- $\displaystyle \int \frac{\ln x}{x}\,dx$: let $u = \ln x$. The answer is $\displaystyle \frac{(\ln x)^2}{2} + C$.
- Always differentiate your answer to confirm.
Practice
Which substitution works best for $\displaystyle \int x^2 e^{x^3}\,dx$?
The derivative of $x^3$ is $3x^2$, which matches the other factor up to a constant.
Find $\displaystyle \int \frac{3x^2}{x^3 + 5}\,dx$.
The top is the derivative of the bottom, so use $\displaystyle \int \frac{du}{u} = \ln|u|$.
Find $\displaystyle \int \frac{\ln x}{x}\,dx$.
With $u = \ln x$, $\displaystyle du = \frac{1}{x}\,dx$, so $\displaystyle \int u\,du = \frac{u^2}{2}$.
Ready for more? Take the Unit 8 test.