Topic 6.9 Integrating Using Substitution: integrate composite functions by reversing the chain rule with u-substitution, including changing limits for definite integrals.

What this topic is asking

The College Board (Topic 6.9) introduces u-substitution, the integration technique that reverses the chain rule. When an integrand contains a composite function times (a constant multiple of) the derivative of the inner function, substituting $u$ for the inner function collapses it to a basic integral.

The method

A worked indefinite substitution

Changing limits for definite integrals

For a definite integral, you have two clean options. Either back-substitute to $x$ and use the original limits, or, more efficiently, change the limits to $u$-values and never return to $x$. If $u = g(x)$, the new limits are $u = g(a)$ and $u = g(b)$. The exam rewards showing this change explicitly. The frequent error is changing the integrand to $u$ but leaving the original $x$-limits in place, then evaluating as if they were $u$-limits. Always either convert the limits to $u$ or convert back to $x$ before using the $x$-limits; never mix.

Recognizing when substitution fits

The skill is recognition: spotting that an integrand is a composite times the derivative of its inside. The classic patterns are $\int (\text{stuff})^n \cdot (\text{derivative of stuff})\,dx$, $\int e^{(\text{stuff})}\cdot(\text{derivative of stuff})\,dx$, and $\int \frac{(\text{derivative of stuff})}{(\text{stuff})}\,dx$ giving a logarithm. When the derivative factor is present only up to a constant, you fix it by factoring the constant in or out (as in the $\frac{1}{3}$ above). When no derivative-of-inside factor is present at all and cannot be supplied by a constant, substitution does not apply, and on the AB exam the integral is then either a basic form or beyond scope. Choosing $u$ to be the inner function of the most complicated part is the reliable first guess.

A worked definite-integral substitution

Why the logarithm pattern appears so often

The pattern $\int \frac{g'(x)}{g(x)}\,dx = \ln|g(x)| + C$ is worth recognizing on sight, because it is the substitution $u = g(x)$ with the numerator supplying $du$. Many AB integrals are built to fit it: a fraction whose numerator is (a constant times) the derivative of its denominator integrates to a logarithm of the denominator. Spotting this saves the full substitution write-up and explains why so many answers in this topic involve $\ln$. The same recognition applies to the exponential pattern $\int g'(x)e^{g(x)}\,dx = e^{g(x)} + C$. Training your eye to see the inside function and its derivative together is what makes substitution fast rather than mechanical.