1. Integration by Parts
Integration by parts is used when two functions are multiplied together.
It comes from the product rule of differentiation.
\[
\int u\frac{dv}{dx}\,dx = uv - \int v\frac{du}{dx}\,dx
\]
Choosing \(u\)
Usually choose logarithmic, inverse trigonometric, algebraic, trigonometric,
then exponential functions in that order.
LIATE Rule
L = Logarithmic
I = Inverse Trigonometric
A = Algebraic
T = Trigonometric
E = Exponential
Worked Example 1
Evaluate:
\[
\int x\cos x\,dx
\]
Step 1 – Choose \(u\) and \(dv\)
\[
u=x
\]
\[
dv=\cos x\,dx
\]
Step 2 – Differentiate and Integrate
\[
\frac{du}{dx}=1
\]
\[
v=\sin x
\]
Step 3 – Apply Formula
\[
\int x\cos x\,dx = x\sin x - \int \sin x\,dx
\]
\[
=x\sin x + \cos x + c
\]
\[
\boxed{x\sin x+\cos x+c}
\]
Worked Example 2
Evaluate:
\[
\int x e^x\,dx
\]
Choose Functions
\[
u=x
\]
\[
dv=e^x\,dx
\]
Differentiate and Integrate
\[
du=dx
\]
\[
v=e^x
\]
Apply Formula
\[
\int xe^x\,dx = xe^x - \int e^x\,dx
\]
\[
=xe^x-e^x+c
\]
\[
=e^x(x-1)+c
\]
\[
\boxed{e^x(x-1)+c}
\]
2. Trigonometric Integration
Some trigonometric integrals are direct while others require identities.
Basic Integrals
\[
\int \sin x\,dx = -\cos x+c
\]
\[
\int \cos x\,dx = \sin x+c
\]
\[
\int \sec^2x\,dx = \tan x+c
\]
\[
\int \csc^2x\,dx = -\cot x+c
\]
Worked Example 3
Evaluate:
\[
\int (3\sin x+4\cos x)\,dx
\]
Integrate Each Term
\[
=3\int \sin x\,dx + 4\int \cos x\,dx
\]
\[
=3(-\cos x)+4(\sin x)+c
\]
\[
\boxed{-3\cos x+4\sin x+c}
\]
Worked Example 4 – Trigonometric Identity
Evaluate:
\[
\int \sin^2x\,dx
\]
Use Identity
\[
\sin^2x=\frac{1-\cos2x}{2}
\]
Substitute
\[
=\int \frac{1-\cos2x}{2}\,dx
\]
\[
=\frac{1}{2}\int 1\,dx - \frac{1}{2}\int \cos2x\,dx
\]
\[
=\frac{x}{2}-\frac{\sin2x}{4}+c
\]
\[
\boxed{\frac{x}{2}-\frac{\sin2x}{4}+c}
\]
3. Exponential Integration
Basic Rule
\[
\int e^x\,dx=e^x+c
\]
\[
\int a^x\,dx=\frac{a^x}{\ln a}+c
\]
Example
\[
\int 2^x\,dx
\]
\[
=\frac{2^x}{\ln2}+c
\]
4. Definite Integration Example
Evaluate:
\[
\int_0^\pi \sin x\,dx
\]
Integrate
\[
=[-\cos x]_0^\pi
\]
Substitute Limits
\[
=[-\cos\pi]-[-\cos0]
\]
\[
=1-(-1)
\]
\[
\boxed{2}
\]
Practice Questions
- \(\int x\sin x\,dx\)
- \(\int x^2e^x\,dx\)
- \(\int \cos^2x\,dx\)
- \(\int (5\sec^2x-2\sin x)\,dx\)
- \(\int_1^2 3x^2\,dx\)
- \(\int x\ln x\,dx\)