Antiderivatives & Standard Integrals: Rules, Techniques & Applications

TAKEAWAYS:

• Antiderivatives are the reverse of derivatives: if $F^{\prime }(x)=f(x)$, then $F(x)$ is the antiderivative of $f(x)$.
• Standard integrals (like $\int x^{n}dx$, $\int \sin xdx$) follow predictable rules and are memorized.
• Substitution method simplifies integrals by changing variables (e.g., $u=1+x^2$).
• Integration by parts uses $\int udv=uv-\int vdu$ for products of functions.
• Applications include finding areas under curves, volumes of solids, and solving real-world problems.

1. What is an Antiderivative?

An antiderivative (or indefinite integral) of a function $f(x)$ is another function $F(x)$ such that: $$F^{\prime }(x)=f(x)$$

• Notation: $\int f(x)dx=F(x)+C$, where $C$ is the constant of integration (since derivatives of constants are zero).
• Example: If $f(x)=2x$, then $F(x)=x^2+C$ because $\frac{d}{dx}(x^2+C)=2x$.

2. Basic Rules of Antiderivatives

Here are the standard integral formulas you must memorize:

Example 1: Find $\int (3x^2+4x+5)dx$. Solution: $$\int (3x^2+4x+5)dx=3\cdot \frac{x^{2+1}}{2+1}+4\cdot \frac{x^{1+1}}{1+1}+5x+C=x^3+2x^2+5x+C$$

3. Integration by Substitution (Reverse Chain Rule)

If $u=g(x)$, then: $$\int f(g(x))\cdot g^{\prime }(x)dx=\int f(u)du$$ Steps:

1. Let $u=$ (a function of $x$).
2. Compute $du=\frac{du}{dx}dx$.
3. Rewrite the integral in terms of $u$.
4. Integrate with respect to $u$.
5. Substitute back $u=g(x)$.

Example 2: Find $\int 2x{e}^{x^2}dx$. Solution:

1. Let $u=x^2$, then $du=2xdx$.
2. The integral becomes $\int e^{u}du$.
3. Integrate: $e^u+C$.
4. Substitute back: $e^{x^2}+C$.

4. Integration by Parts

For integrals of the form $\int udv$, use: $$\int udv=uv-\int vdu$$ LIATE Rule (for choosing $u$):

• Logarithmic functions
• Inverse trigonometric functions
• Algebraic functions
• Trigonometric functions
• Exponential functions

Example 3: Find $\int x\ln xdx$. Solution:

1. Let $u=\ln x$ (logarithmic), $dv=xdx$.
2. Then $du=\frac{1}{x}dx$, $v=\frac{x^2}{2}$.
3. Apply the formula: $$\int x\ln xdx=\frac{x^2}{2}\ln x-\int \frac{x^2}{2}\cdot \frac{1}{x}dx=\frac{x^2}{2}\ln x-\frac{1}{2}\int xdx$$
4. Integrate the remaining term: $$=\frac{x^2}{2}\ln x-\frac{x^2}{4}+C$$

flowchart TD
    A["Choose u and dv"] --> B["Compute du and v"]
    B --> C["Apply: \( \int u \, dv = uv - \int v \, du \)"]
    C --> D["Simplify and integrate"]
    D --> E["Final answer: \( \frac{x^2}{2} \ln x - \frac{x^2}{4} + C \)"]

5. Applications of Antiderivatives

(a) Finding Areas Under Curves

The area under $y=f(x)$ from $x=a$ to $x=b$ is: $$\text{Area}={\int }_a^{b}f(x)dx=F(b)-F(a)$$ Example 4: Find the area under $y=x^2$ from $x=1$ to $x=3$. Solution: $${\int }_1^3{x}^{2}dx={\left[\frac{x^3}{3}\right]}_1^3=\frac{27}{3}-\frac{1}{3}=9-\frac{1}{3}=\frac{26}{3}$$

(b) Solving Differential Equations

Antiderivatives help solve equations like: $$\frac{dy}{dx}=f(x)$$ Example 5: Solve $\frac{dy}{dx}=3x^2$. Solution: $$y=\int 3x^{2}dx=x^3+C$$

6. Common Mistakes to Avoid

1. Forgetting the constant $+C$ in indefinite integrals.
2. Incorrect substitution (e.g., missing $du$ or $dx$).
3. Misapplying integration by parts (wrong choice of $u$ and $dv$).
4. Skipping limits in definite integrals (always evaluate $F(b)-F(a)$).

7. NEB Board-Style Questions

Short Answer (5 marks)

1. Find $\int (4x^3-2x+1)dx$.
2. Evaluate ${\int }_0^2(3x^2+2x)dx$.
3. Use substitution to find $\int \frac{x}{\sqrt{1+x^2}}dx$.

Long Answer (10 marks)

4. Find $\int x{e}^{x}dx$ using integration by parts.
5. Compute the area under $y=\sin x$ from $x=0$ to $x=\pi$.
6. Solve the differential equation $\frac{dy}{dx}=\cos x$.

Problem-Solving (15 marks)

7. A particle moves with velocity $v(t)=3t^2+2t$. Find its displacement from $t=0$ to $t=2$.
8. Evaluate $\int \frac{1}{x\ln x}dx$ using substitution.

Exam Tip

• Memorize standard integrals (they save time in exams).
• Practice substitution and parts—these are high-yield topics.
• Always check units (e.g., area under a curve should be in square units).
• Show all steps—partial credit is given for correct methods.
• For definite integrals, evaluate at the bounds before adding $C$.