MATH1231 - Techniques of Integration

Substitution

$\int_a^bf(g(x)) \cdot g'(x)dx = \int_{g(a)}^{g(b)}f(x)dx$

i.e.

$\int f(u) \frac {du}{dx}dx = \int f(u)du$

Parts

$\int_a^b f(x) \cdot g'(x)dx = \left[ f(x)g(x) \right]_{a}^{b} - \int_a^b f'(x)g(x)dx$

Trig Integrals

Try and get them in terms of $cos^n(x) \cdot sin(x)$ or $sin^n(x)cos(x)$, and then use substitution.

$\int cos^m(x)sin^n(x)dx$ (At least one of m or n is Odd)

m odd → use u = sin(x)
n odd → use u = cos(x)
then use $sin^2(x) + cos^2(x) = 1$

$\int cos^m(x)sin^n(x)dx$ (Both m and n are even)

Use

$cos^2(x) = \frac {1 + cos(2x)}{2} and sin^2(x) = \frac {1- cos2(x)}{2}$

$\int cos(2x)^2dx = \frac {1}{2} \int 1+cos(4x)dx$

$\int cos(mx)sin(nx)dx$ or $\int cos(mx)cos(nx)dx$ or $\int sin(mx)sin(nx)dx$

Use

(1)
\begin{align} \begin {array} {r l} sinAcosB & = ½(sin(A-B) + sin(A+B)) \\ cosAcosB & = ½(cos(A-B) + cos(A+B)) \\ sinAsinB & = ½(cos(A-B) – cos(A+B)) \\ \end {array} \end{align}

Powers of tan(x) and sec(x)

Use

(2)
\begin{align} \begin {array} {r l} tan^2(x) + 1 & = sec^2(x) \\ \frac {d}{dx} (tan(x)) & = sec^2(x) \\ \frac {d}{dx} (sec(x)) & = sec(x)tan(x) \\ \end {array} \end{align}