Antiderivatives: Definition, Rules, Integration Techniques & Applications

What is an Antiderivative?

An antiderivative (or indefinite integral) is the reverse process of differentiation. If $F^{\prime }(x)=f(x)$, then $F(x)$ is the antiderivative of $f(x)$. We write this as: $$\int f(x)dx=F(x)+C$$ where $C$ is the constant of integration (since differentiating a constant gives zero).

Key Idea:

• Differentiation finds the slope of a curve.
• Antiderivative finds the original curve given its slope.

Basic Rules of Antiderivatives

1. Power Rule

For any real number $n\ne -1$: $$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$$ Example: Find $\int 3x^{4}dx$. Solution: $$\int 3x^{4}dx=3\cdot \frac{x^{4+1}}{4+1}+C=\frac{3x^5}{5}+C$$

2. Constant Multiple Rule

$$\int k\cdot f(x)dx=k\int f(x)dx$$ Example: Find $\int 5dx$. Solution: $$\int 5dx=5x+C$$

3. Sum/Difference Rule

$$\int [f(x)\pm g(x)]dx=\int f(x)dx\pm \int g(x)dx$$ Example: Find $\int (2x^3+4x)dx$. Solution: $$\int 2x^{3}dx+\int 4xdx=\frac{2x^4}{4}+\frac{4x^2}{2}+C=\frac{x^4}{2}+2x^2+C$$

Special Cases

1. Exponential Function

$$\int e^{x}dx=e^x+C$$

2. Trigonometric Functions

$$\int \sin xdx=-\cos x+C$$ $$\int \cos xdx=\sin x+C$$

3. Reciprocal of $x$

$$\int \frac{1}{x}dx=\ln |x|+C$$

Solved Examples

Example 1: Power Rule

Find $\int (4x^3-2x+1)dx$. Solution: $$\int 4x^{3}dx-\int 2xdx+\int 1dx=x^4-x^2+x+C$$

Example 2: Mixed Functions

Find $\int (e^x+\sin x)dx$. Solution: $$\int e^{x}dx+\int \sin xdx=e^x-\cos x+C$$

Geometric Interpretation

The antiderivative represents the area under the curve $f(x)$ from a starting point to $x$. The constant $C$ shifts the curve vertically.

Applications

1. Physics: Finding displacement from velocity.
2. Economics: Calculating total cost from marginal cost.
3. Biology: Modeling population growth.

Common Mistakes to Avoid

• Forgetting the constant $C$.
• Misapplying the power rule (e.g., $\int \frac{1}{x}dx\ne \frac{x^0}{0}+C$).
• Incorrectly integrating trigonometric functions (e.g., $\int \cos xdx\ne -\sin x+C$).

NEB Board-Style Questions

Short Answer (5 marks each)

1. Find $\int (3x^2-2x+5)dx$.
2. Evaluate $\int \frac{1}{x^2}dx$.
3. If $F^{\prime }(x)=4x^3+2$, find $F(x)$.

Long Answer (10 marks)

4. Find the antiderivative of $f(x)=e^x+\sin x-\frac{1}{x}$. Verify by differentiating.
5. A particle moves with velocity $v(t)=3t^2+2t$. Find its displacement from $t=0$ to $t=2$.

Exam Tip

• Always include $+C$ in your final answer.
• Practice differentiation first—antiderivatives are its reverse.
• Check your work by differentiating the result.
• Memorize basic rules (power, exponential, trigonometric).
• Break complex integrals into simpler parts using sum/difference rules.

Visual Summary:

graph TD
    A["Differentiation"] -->|"Reverse"| B["Antiderivative"]
    B --> C["Power Rule"]
    B --> D["Exponential Rule"]
    B --> E["Trigonometric Rule"]
    B --> F["Reciprocal Rule"]
    C --> G["Add 1 to exponent"]
    D --> H["Integral of e^x is e^x"]
    E --> I["Integral of sin x is -cos x"]
    F --> J["Integral of 1/x is ln|x|"]