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Integral Calculus

Chapter 2 · Techniques

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Substitution (reverse chain rule)
∫f(g(x))g′(x) dx=∫f(u) du\int f(g(x))g'(x)\,dx = \int f(u)\,du
Substitution steps
Let uu = inner function; replace g′(x)dxg'(x)dx with dudu; integrate in uu; substitute back.
By parts (reverse product rule)
∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du
Choosing u (LIATE)
Log, Inverse-trig, Algebraic, Trig, Exponential — pick uu earliest in this list.
Common results
∫xexdx=(x−1)ex+C,∫xcos⁡x dx=xsin⁡x+cos⁡x+C\int xe^x dx = (x-1)e^x + C,\quad \int x\cos x\,dx = x\sin x + \cos x + C
Substitution
∫f(g(x))g′(x) dx=∫f(u) du\int f(g(x))g'(x)\,dx = \int f(u)\,du
Change limits too for definite integrals.
By parts
∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du
Pick u by LIATE (log, inverse-trig, algebraic, trig, exp).
Common trap
After substitution, convert limits to u (or back-substitute before applying x-limits).