U-Substitution

Integration by Substitution (a.k.a. "u-Substitution)

It's like "reverse chain rule" for integration.

Identifying Clues

• If you see a function wrapped up in another function, and its derivative shows up as well
• If you see both a $\sin$ and a $\cos$
• If you see a $\ln x$ and some $x$ in the denominator
• If you see a $\sqrt{x}$ or $\frac{1}{x}$ wrapped inside another function

Strategy

1. Pick one part of the function and let that be $u$

• $u'$ should appear elsewhere in the function
• Pick $u$ as the function wrapped inside another
• Pick $u$ as $\ln x$
• Pick $u$ as the denominator
• Sometimes we need to factor or complete the square before we do our u-substitution

2. Differentiate $u$ → $du=...dx$

3. Solve for $dx$

4. Substitute $u$ , replace $dx$

• For definite integrals, we also want to replace the upper and lower integral bounds with u-bounds
• If you still have x's remaining in your integral after substitution, go back to your $u=...$ formula and do another substitution

5. Integrate the new function with respect to $u$

6. Subsitute the original $x$ expression back in for $u$

Example

Evaluate $\displaystyle\int_{ }^{ }\frac{2x}{\sqrt{x^2-1}}dx$