Integration Techniques, Fundamental Theorem, and Applications

1. Introduction to Integration

1.1 Definition of an Integral

• Indefinite Integral: The antiderivative of a function $f(x)$, denoted as: $$\int f(x)dx=F(x)+C\text{where}{F}^{\prime }(x)=f(x).$$
• Definite Integral: The signed area under $f(x)$ from $a$ to $b$, written as: $${\int }_a^{b}f(x)dx=F(b)-F(a).$$

How it works:

• Integration reverses differentiation. For example, if $F^{\prime }(x)=2x$, then $\int 2xdx=x^2+C$.
• Geometric interpretation: Definite integrals compute net area between $f(x)$ and the x-axis (positive above, negative below).

Worked Example: Find $\int (3x^2+4x+5)dx$. Solution: $$\int (3x^2+4x+5)dx=x^3+2x^2+5x+C.$$

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2. Basic Integration Rules

2.1 Power Rule

For $n\ne -1$: $$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C.$$ Example: $\int x^{3}dx=\frac{x^4}{4}+C$.

2.2 Exponential and Trigonometric Integrals

Worked Example: Find $\int (e^{2x}+\sin 3x)dx$. Solution: $$\int e^{2x}dx=\frac{e^{2x}}{2}+C_1,\int \sin 3xdx=-\frac{\cos 3x}{3}+C_2.$$ Combined: $$\frac{e^{2x}}{2}-\frac{\cos 3x}{3}+C.$$

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3. Integration Techniques

3.1 Substitution Method (u-substitution)

How it works: Replace $u=g(x)$, then $du=g^{\prime }(x)dx$. Use when the integrand is a composite function. Formula: $$\int f(g(x))g^{\prime }(x)dx=\int f(u)du.$$

Worked Example: Find $\int 2x{e}^{x^2}dx$. Solution: Let $u=x^2$, $du=2xdx$. $$\int e^{x^2}\cdot 2xdx=\int e^{u}du=e^u+C=e^{x^2}+C.$$

3.2 Integration by Parts

How it works: Derived from the product rule: $$\int udv=uv-\int vdu.$$ Choosing $u$ and $dv$: Use LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential).

Worked Example: Find $\int x\ln xdx$. Solution: Let $u=\ln x$, $dv=xdx$. Then $du=\frac{1}{x}dx$, $v=\frac{x^2}{2}$. $$\int x\ln xdx=\frac{x^2}{2}\ln x-\int \frac{x^2}{2}\cdot \frac{1}{x}dx=\frac{x^2}{2}\ln x-\frac{x^2}{4}+C.$$

3.3 Partial Fractions

How it works: Decompose rational functions $\frac{P(x)}{Q(x)}$ where $\mathrm{deg}P<\mathrm{deg}Q$ into simpler fractions. Steps:

1. Factor $Q(x)$ into linear/irreducible quadratic factors.
2. Write as sum of partial fractions.
3. Solve for coefficients.

Worked Example: Find $\int \frac{3x+5}{x^2-x-2}dx$. Solution: Factor denominator: $x^2-x-2=(x-2)(x+1)$. Assume: $$\frac{3x+5}{(x-2)(x+1)}=\frac{A}{x-2}+\frac{B}{x+1}.$$ Solve for $A$ and $B$: $$3x+5=A(x+1)+B(x-2).$$ Let $x=2$: $11=3A\Rightarrow A=\frac{11}{3}$. Let $x=-1$: $2=-3B\Rightarrow B=-\frac{2}{3}$. Now integrate: $$\int \left(\frac{11/3}{x-2}-\frac{2/3}{x+1}\right)dx=\frac{11}{3}\ln |x-2|-\frac{2}{3}\ln |x+1|+C.$$

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4. Fundamental Theorem of Calculus

4.1 First Fundamental Theorem

If $f$ is continuous on $[a,b]$ and $F(x)={\int }_a^{x}f(t)dt$, then: $$F^{\prime }(x)=f(x).$$ Implication: Differentiation and integration are inverse operations.

4.2 Second Fundamental Theorem

If $f$ is continuous on $[a,b]$, then: $${\int }_a^{b}f(x)dx=F(b)-F(a),$$ where $F$ is any antiderivative of $f$.

Worked Example: Evaluate ${\int }_1^3(2x+1)dx$. Solution: Find antiderivative: $F(x)=x^2+x$. Apply theorem: $$F(3)-F(1)=(9+3)-(1+1)=10.$$

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5. Applications of Integration

5.1 Area Under a Curve

The area $A$ between $f(x)$, the x-axis, and $x=a$ to $x=b$ is: $$A={\int }_a^b|f(x)|dx.$$ Worked Example: Find the area between $f(x)=x^2-4$ and the x-axis from $x=-2$ to $x=2$. Solution: Find roots: $x^2-4=0\Rightarrow x=\pm 2$. Split integral: $$A={\int }_{-2}^2|x^2-4|dx=2{\int }_0^2(4-x^2)dx=2{\left[4x-\frac{x^3}{3}\right]}_0^2=\frac{16}{3}.$$

5.2 Volume of Revolution

Disk Method: For $y=f(x)$ rotated about the x-axis: $$V=\pi {\int }_a^b[f(x){]}^{2}dx.$$ Worked Example: Find the volume obtained by rotating $y=\sqrt{x}$ about the x-axis from $x=0$ to $x=4$. Solution: $$V=\pi {\int }_0^4(\sqrt{x}{)}^{2}dx=\pi {\int }_0^{4}xdx=\pi {\left[\frac{x^2}{2}\right]}_0^4=8\pi .$$

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6. Comparison of Integration Techniques

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7. Common Mistakes and Pitfalls

1. Forgetting $+C$: Indefinite integrals always include the constant of integration.
2. Incorrect Substitution: Ensure $du$ matches the differential in the integrand.
3. Partial Fractions Errors: Always factor the denominator completely.
4. Sign Errors: Pay attention to absolute values in area problems.
5. Misapplying Limits: For definite integrals, ensure correct bounds and antiderivative evaluation.

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8. Exam Tip

What to Expect in TU Exams

1. Theory Questions (20%):

   • Define indefinite/definite integrals, Fundamental Theorem of Calculus.
   • State conditions for integrability (e.g., continuity).
   • Example Question: "State the Fundamental Theorem of Calculus and explain its significance."

2. Problem-Solving (80%):

   • Direct Integration: Apply power/exponential rules (5-10 marks).
   • Techniques: Substitution, parts, or partial fractions (10-15 marks).
     • Example: "Evaluate $\int \frac{x^2+1}{x^3+x}dx$ using partial fractions."
   • Applications: Area/volume problems (10-15 marks).
     • Example: "Find the area bounded by $y=x^2$, $y=4$, and the y-axis."
   • Definite Integrals: Combine techniques with limits (5-10 marks).
     • Example: "Compute ${\int }_0^{\pi /2}x\sin xdx$ using integration by parts."

3. Short Answer (Multiple Choice/True-False):

   • Identify correct antiderivatives or integration techniques.
   • Example: "Which method would you use for $\int x^2{e}^{3x}dx$?" Options: A) Substitution, B) Integration by parts, C) Partial fractions.

Marking Scheme Insights

• Full Marks (10/10): Correct method, accurate calculations, proper limits, and clear steps.
• Partial Marks (5/10): Correct approach but arithmetic errors or missing $+C$.
• Zero Marks: Incorrect technique (e.g., using substitution where parts are needed).

Recommended Practice

1. Drill Basic Integrals: Master power, exponential, and trigonometric rules.
2. Mixed Techniques: Solve problems requiring substitution + parts (e.g., $\int x{e}^{x^2}\sin xdx$).
3. Application Problems: Focus on area/volume with piecewise functions or absolute values.
4. Past Papers: TU often repeats question patterns from 2018-2023 (e.g., partial fractions with quadratic denominators).

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