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Integration techniques
Elementary integrals
\int \frac{1}{a^2 + x^2} dx = \frac{1}{a} \arctan(\frac{x}{a}) + C
\int \frac{1}{\sqrt{a^2-x^2}} dx = \arcsin(\frac{x}{a}) + C
Linearity of the integral
\int Af(x) + Bg(x)dx = A\int f(x)dx + B\int g(x)dx
Proof: is missing.
Substitution
Suppose that g is a differentiable on [a,b], that satisfies g(a)=A and g(b)=B. Also suppose that f is continuous on the range of g, then
let u = g(x) then du = g'(x)dx,
\int_a^b f(g(x))g'(x)dx = \int_A^B f(u)du.
Inverse substitution
Inverse substitutions appear to make the integral more complicated, thereby this strategy must act as last resort. Substituting x=g(u) in the integral
\int_a^b f(x)dx,
leads to the integral
\int_{x=a}^{x=b} f(g(u))g'(u)du.
Integration by parts
Suppose U(x) and V(x) are two differentiable functions. According to the product rule,
\frac{d}{dx}(U(x)V(x)) = U(x) \frac{dV}{dx} + V(x) \frac{dU}{dx}.
Integrating both sides of this equation and transposing terms
\int U(x) \frac{dV}{dx} dx = U(x)V(x) - \int V(x) \frac{dU}{dx} dx,
obtaining:
\int U dV = U V - \int V dU.
For definite integrals that is:
\int_a^b f'(x)g(x)dx = [f(x)g(x)]_a^b - \int_a^b f(x)g'(x)dx.
Integration of rational functions
Let P(x) and Q(x) be polynomial functions with real coefficients. Forming a rational function, \frac{P(x)}{Q(x)}. Let \frac{P(x)}{Q(x)} be a strictly proper rational function, that is; \mathrm{deg}(P(x)) < \mathrm{deg}(Q(x)). If the function is not it can be possibly made into a strictly proper rational function by using long division.
Then, Q(x) can be factored into the product of a constant K, real linear factors of the form x-a_i, and real quadratic factors of the form $x^2+b_ix + c_i having no real roots.
The rational function can be expressed as a sum of partial fractions. Corresponding to each factor (x-a)^m of Q(x) the decomposition contains a sum of fractions of the form
\frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + ... + \frac{A_m}{(x-a)^m}.
Corresponding to each factor (x^2+bx+c)^n of Q(x) the decomposition contains a sum of fractions of the form
\frac{B_1x+C_1}{x^2+bx+c} + \frac{B_2x+C_2}{(x^2+bx+c)^2} + ... + \frac{B_nx+C_n}{(x^2+bx+c)^n}.
The constant A_1,A_2,...,A_m,B_1,B_2,...,B_n,C_1,C_2,....,C_n can be determined by adding up the fractions in the decomposition and equating the coefficients of like powers of x in the numerator of the sum those in P(x).