Revise: Methods of Integration
Basic integration, the substitution method, integration by parts, and the partial fraction method.
∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n≠−1); integrate term by term.
∫3x²dx = x³+C.
∫eˣdx=eˣ+C; ∫sin(x)dx=−cos(x)+C; ∫cos(x)dx=sin(x)+C.
∫4cos(x)dx = 4sin(x)+C.
Let u=g(x) when g'(x) also appears in the integrand.
∫2x(x²+1)³dx = (x²+1)⁴/4+C.
(px+q)/[(x−a)(x−b)] = A/(x−a) + B/(x−b).
Substitute x=a and x=b to isolate A and B.
∫A/(x−a) dx = A·ln|x−a| + C.
∫1/[(x−1)(x+2)]dx = (1/3)ln|x−1|−(1/3)ln|x+2|+C.
Check any integral by differentiating the answer.
∫₀¹x²dx = 1/3.
Easy to swap the sign between ∫sin and ∫cos — double check.
∫2eˣ+3sin(x)dx = 2eˣ−3cos(x)+C.
Comes from reversing the Product Rule; pick u to simplify under differentiation.
∫x cos(x)dx = x sin(x)+cos(x)+C.
Only applies directly to distinct linear factors.
1/[(x−1)(x+1)]: A=1/2, B=−1/2.
Never forget the absolute value bars inside the logarithm.
∫1/[(x−1)(x+1)]dx = (1/2)ln|x−1|−(1/2)ln|x+1|+C.