What is Integration?
Integration is the reverse process of differentiation. It is used to find functions,
areas under curves, displacement from velocity, and many applications in mathematics and physics.
\[
\int f(x)\,dx
\]
The symbol \(\int\) means integrate, while \(dx\) shows the variable with respect to which we integrate.
Basic Integration Rules
Rule 1 – Power Rule
\[
\int x^n\,dx = \frac{x^{n+1}}{n+1}+c
\quad \text{where } n\neq -1
\]
Add 1 to the power, then divide by the new power.
Rule 2 – Constant Rule
\[
\int k\,dx = kx+c
\]
Rule 3 – Constant Multiple Rule
\[
\int kf(x)\,dx = k\int f(x)\,dx
\]
Important Note
\[
+c
\]
Always add the constant of integration because differentiation removes constants.
Worked Example 1
Find:
\[
\int 6x^2\,dx
\]
Step 1 – Apply Constant Rule
\[
=6\int x^2\,dx
\]
Step 2 – Use Power Rule
\[
=6\left(\frac{x^{2+1}}{2+1}\right)
\]
\[
=6\left(\frac{x^3}{3}\right)
\]
Step 3 – Simplify
\[
=2x^3+c
\]
\[
\boxed{\int 6x^2\,dx = 2x^3+c}
\]
Worked Example 2
Find:
\[
\int (3x^4-2x+7)\,dx
\]
Step 1 – Integrate Each Term Separately
\[
=\int 3x^4\,dx - \int 2x\,dx + \int 7\,dx
\]
Step 2 – Integrate
\[
=3\left(\frac{x^5}{5}\right)-2\left(\frac{x^2}{2}\right)+7x+c
\]
Step 3 – Simplify
\[
=\frac{3x^5}{5}-x^2+7x+c
\]
\[
\boxed{\frac{3x^5}{5}-x^2+7x+c}
\]
Definite Integration
Definite integration is used to calculate the area under a curve between two limits.
\[
\int_a^b f(x)\,dx
\]
Formula
\[
\int_a^b f(x)\,dx = F(b)-F(a)
\]
Example
\[
\int_1^3 2x\,dx
\]
Integrate first:
\[
=\left[x^2\right]_1^3
\]
Substitute limits:
\[
=3^2-1^2
\]
\[
=9-1
\]
\[
\boxed{8}
\]
Common Mistakes in Integration
- Forgetting to add the constant \(+c\)
- Adding instead of dividing by the new power
- Not increasing the power first
- Using the power rule when the power is \(-1\)
- Wrong substitution of limits in definite integration
Practice Questions
- \(\int 5x^3\,dx\)
- \(\int (2x^2+4x-1)\,dx\)
- \(\int_0^2 x^2\,dx\)
- \(\int (7x^{-2})\,dx\)
- \(\int (4-3x^5)\,dx\)