Question

Calculus people! How do I know when to use the FTC or U-substitution when evaluating integrals?

Answer 1

Use the FTC when you're asked to take the derivative of an integral, as in
\[\frac{d}{dx}\int_c^{g(x)}f(u)~du\stackrel{FTC}{=}f(g(x)) \times g'(x)\]
and also when evaluating definite integrals with simple integrands:
\[\int_a^b f(x)~dx=F(b)-F(a)\]
when \(F(x)\) is the antiderivative of \(f(x)\).

You use substitutions as a method of rewriting integrals in terms that are more approachable. For instance, the integral
\[\int 2x\sqrt{x^2+1}~dx\]
doesn't look readily solvable, but if we make the substitution \(u=x^2+1\) and take the differential \(du=2x~dx\), we get
\[\int\color{blue}{2x}\sqrt{\color{red}{x^2+1}}~\color{blue}{dx}=\int\sqrt{\color{red}{ u}}~\color{blue}{du}\]
which can then be integrated using the power rule.