What You Should Expect in Standard Integrals
- Recognise the form of the integral immediately
- Know the standard integration formulas
- Apply correct algebraic simplification
- Always add the constant of integration \(+c\)
- Use identities where necessary
- Handle trigonometric and exponential forms confidently
1. Power Rule Integrals
\[
\int x^n\,dx=\frac{x^{n+1}}{n+1}+c
\quad n\neq -1
\]
Example 1
Evaluate:
\[
\int x^4\,dx
\]
Add 1 to the power:
\[
4+1=5
\]
Divide by the new power:
\[
=\frac{x^5}{5}+c
\]
\[
\boxed{\frac{x^5}{5}+c}
\]
Example 2
Evaluate:
\[
\int 7x^3\,dx
\]
Take out the constant:
\[
=7\int x^3\,dx
\]
\[
=7\left(\frac{x^4}{4}\right)+c
\]
\[
\boxed{\frac{7x^4}{4}+c}
\]
2. Integrals with Negative Powers
\[
\int x^{-n}\,dx
\]
Example 1
\[
\int x^{-2}\,dx
\]
Add 1 to the power:
\[
-2+1=-1
\]
\[
=\frac{x^{-1}}{-1}+c
\]
\[
=-\frac{1}{x}+c
\]
\[
\boxed{-\frac{1}{x}+c}
\]
Example 2
\[
\int 5x^{-3}\,dx
\]
\[
=5\left(\frac{x^{-2}}{-2}\right)+c
\]
\[
=-\frac{5}{2x^2}+c
\]
\[
\boxed{-\frac{5}{2x^2}+c}
\]
3. Special Logarithmic Integral
\[
\int \frac{1}{x}\,dx=\ln|x|+c
\]
Example 1
\[
\int \frac{1}{x}\,dx
\]
\[
\boxed{\ln|x|+c}
\]
Example 2
\[
\int \frac{3}{x}\,dx
\]
\[
=3\ln|x|+c
\]
\[
\boxed{3\ln|x|+c}
\]
4. Exponential Integrals
\[
\int e^x\,dx=e^x+c
\]
\[
\int a^x\,dx=\frac{a^x}{\ln a}+c
\]
Example 1
\[
\int e^x\,dx
\]
\[
\boxed{e^x+c}
\]
Example 2
\[
\int 2^x\,dx
\]
\[
=\frac{2^x}{\ln2}+c
\]
\[
\boxed{\frac{2^x}{\ln2}+c}
\]
5. Trigonometric Integrals
\[
\int \sin x\,dx=-\cos x+c
\]
\[
\int \cos x\,dx=\sin x+c
\]
\[
\int \sec^2x\,dx=\tan x+c
\]
Example 1
\[
\int \sin x\,dx
\]
\[
\boxed{-\cos x+c}
\]
Example 2
\[
\int (2\cos x+\sec^2x)\,dx
\]
\[
=2\sin x+\tan x+c
\]
\[
\boxed{2\sin x+\tan x+c}
\]
6. Integration by Parts
\[
\int u\frac{dv}{dx}\,dx=uv-\int v\frac{du}{dx}\,dx
\]
Example 1
\[
\int x\cos x\,dx
\]
\[
=x\sin x-\int \sin x\,dx
\]
\[
=x\sin x+\cos x+c
\]
\[
\boxed{x\sin x+\cos x+c}
\]
Example 2
\[
\int xe^x\,dx
\]
\[
=xe^x-e^x+c
\]
\[
=e^x(x-1)+c
\]
\[
\boxed{e^x(x-1)+c}
\]
7. Definite Integrals
\[
\int_a^b f(x)\,dx=F(b)-F(a)
\]
Example 1
\[
\int_0^2 x\,dx
\]
\[
=\left[\frac{x^2}{2}\right]_0^2
\]
\[
=\frac{2^2}{2}-0
\]
\[
\boxed{2}
\]
Example 2
\[
\int_1^3 2x\,dx
\]
\[
=[x^2]_1^3
\]
\[
=9-1
\]
\[
\boxed{8}
\]
Practice Questions
- \(\int 9x^5\,dx\)
- \(\int x^{-4}\,dx\)
- \(\int \frac{5}{x}\,dx\)
- \(\int 3^x\,dx\)
- \(\int (4\sin x+2\cos x)\,dx\)
- \(\int x\sin x\,dx\)
- \(\int_0^1 4x^3\,dx\)