Techniques of Integration: Substitution, Parts, Partial Fractions & Trig

TAKEAWAYS:

• Substitution simplifies integrals by reversing chain rule: replace $u=g(x)$ and adjust $dx$ to $du$.
• Integration by parts uses $\int udv=uv-\int vdu$ to break complex products into simpler terms (LIATE rule: Logs, Inverse trig, Algebraic, Trig, Exponential).
• Partial fractions decomposes $\frac{P(x)}{Q(x)}$ into simpler fractions when $Q(x)$ factors into linear/quadratic terms.
• Trigonometric integrals use identities (e.g., ${\sin }^{2}x=\frac{1-\cos 2x}{2}$) to rewrite integrands before applying substitution.
• Real-world ties: eSewa’s fee calculation uses definite integrals to sum transaction costs; Pathao’s route optimization relies on minimizing integral-based distance functions.
• Exam focus: Show every substitution step and label partial fraction denominators clearly—partial credit is lost for skipped algebra.

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1. Integration by Substitution (Reverse Chain Rule)

Definition: If $u=g(x)$ and $du=g^{\prime }(x)dx$, then $$\int f(g(x))\cdot g^{\prime }(x)dx=\int f(u)du.$$ How it works:

1. Identify $u=g(x)$ such that $du=g^{\prime }(x)dx$ appears in the integrand.
2. Rewrite the integral in terms of $u$.
3. Integrate with respect to $u$, then substitute back $x$.

Worked Example 1: Evaluate $\int 2x{e}^{x^2}dx$. Solution: Let $u=x^2$, then $du=2xdx$. $$\int 2x{e}^{x^2}dx=\int e^{u}du=e^u+C=e^{x^2}+C.$$

Real-World Tie: eSewa Transaction Fees eSewa calculates cumulative fees for bulk transactions using definite integrals. For example, if the fee $F(x)$ for a transaction of size $x$ is $F(x)=0.01x^2$, the total fee for transactions from $x=1$ to $x=100$ is: $${\int }_1^{100}0.01x^{2}dx=0.01{\left[\frac{x^3}{3}\right]}_1^{100}=333.33-0.0033\approx 333.33\text{ NPR}.$$ Here, substitution with $u=x^3$ simplifies the calculation.

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2. Integration by Parts ($\int udv=uv-\int vdu$)

Definition: Derived from the product rule for differentiation. Choose $u$ and $dv$ using the LIATE rule (Logs, Inverse trig, Algebraic, Trig, Exponential).

Worked Example 2: Evaluate $\int x\ln xdx$. Solution: Let $u=\ln x$ (Logs first in LIATE), $dv=xdx$. Then $du=\frac{1}{x}dx$, $v=\frac{x^2}{2}$. Apply the formula: $$\int x\ln xdx=\frac{x^2}{2}\ln x-\int \frac{x^2}{2}\cdot \frac{1}{x}dx=\frac{x^2}{2}\ln x-\frac{1}{2}\int xdx.$$ Simplify: $$=\frac{x^2}{2}\ln x-\frac{x^2}{4}+C.$$

Real-World Tie: Pathao Delivery Route Optimization Pathao uses integration by parts to model the total waiting time for deliveries. If the waiting time $W(t)$ at time $t$ is $W(t)=t\ln t$ (due to traffic congestion), the total waiting time from $t=1$ to $t=T$ is: $${\int }_1^{T}t\ln tdt={\left[\frac{t^2}{2}\ln t-\frac{t^2}{4}\right]}_1^T.$$ This helps Pathao estimate delays and adjust driver routes dynamically.

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3. Partial Fractions for Rational Functions

Definition: Decompose $\frac{P(x)}{Q(x)}$ where $\mathrm{deg}P<\mathrm{deg}Q$ into simpler fractions. Rules:

• Factor $Q(x)$ into linear $(x-a)$ or irreducible quadratic $(x^2+bx+c)$ terms.
• Write partial fractions and solve for constants.

Worked Example 3: Evaluate $\int \frac{3x+5}{x^2-x-6}dx$. Solution: Factor denominator: $x^2-x-6=(x-3)(x+2)$. Assume: $$\frac{3x+5}{(x-3)(x+2)}=\frac{A}{x-3}+\frac{B}{x+2}.$$ Multiply through by $(x-3)(x+2)$: $$3x+5=A(x+2)+B(x-3).$$ Solve for $A$ and $B$:

• Let $x=3$: $9+5=5A\Rightarrow A=2$.
• Let $x=-2$: $-6+5=-5B\Rightarrow B=-1$. Now integrate: $$\int \left(\frac{2}{x-3}-\frac{1}{x+2}\right)dx=2\ln |x-3|-\ln |x+2|+C.$$

Real-World Tie: NTC’s Network Traffic Analysis NTC models data packet delays in networks using rational functions. For example, if the delay $D(x)$ for a packet of size $x$ is: $$D(x)=\frac{2x+1}{x^2-x},$$ partial fractions decompose it into: $$D(x)=\frac{1}{x}+\frac{1}{x-1},$$ allowing NTC to analyze cumulative delays ${\int }_1^{L}D(x)dx$ for packets up to size $L$.

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4. Trigonometric Integrals

Key Identities:

1. ${\sin }^{2}x=\frac{1-\cos 2x}{2}$
2. ${\cos }^{2}x=\frac{1+\cos 2x}{2}$
3. ${\tan }^{2}x={\sec }^{2}x-1$

Worked Example 4: Evaluate $\int {\sin }^{3}xdx$. Solution: Rewrite ${\sin }^{3}x=\sin x(1-{\cos }^{2}x)$: $$\int {\sin }^{3}xdx=\int \sin x(1-{\cos }^{2}x)dx.$$ Let $u=\cos x$, $du=-\sin xdx$: $$=-\int (1-u^2)du=-u+\frac{u^3}{3}+C=-\cos x+\frac{{\cos }^{3}x}{3}+C.$$

Real-World Tie: YouTube’s Video Compression YouTube uses trigonometric integrals to optimize waveform compression. For a signal $f(t)={\sin }^{3}t$, the integral ${\int }_0^T{\sin }^{3}tdt$ represents the total energy of the signal over time $T$. Simplifying this integral (as above) reduces computational load during encoding.

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Comparison Table: Techniques

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5. Solving Past Exam Questions

Question (a): Find the area enclosed by $x^2+y^2=16$.

Solution: Solve for $y$: $y=\pm \sqrt{16-x^2}$. Area $A=4\times {\int }_{-4}^4\sqrt{16-x^2}dx$. Use substitution $x=4\sin \theta$, $dx=4\cos \theta d\theta$: $$A=4\times {\int }_{-\pi /2}^{\pi /2}\sqrt{16-16{\sin }^2\theta }\cdot 4\cos \theta d\theta =64{\int }_{-\pi /2}^{\pi /2}{\cos }^2\theta d\theta .$$ Using ${\cos }^2\theta =\frac{1+\cos 2\theta }{2}$: $$A=64{\left[\frac{\theta }{2}+\frac{\sin 2\theta }{4}\right]}_{-\pi /2}^{\pi /2}=64\times \frac{\pi }{2}=32\pi .$$

Question (b): Evaluate ${\int }_0^1{x}^3\sqrt{1+2x^4}dx$. Solution: Let $u=1+2x^4$, $du=8x^{3}dx\Rightarrow x^{3}dx=\frac{du}{8}$. Change limits: $x=0\Rightarrow u=1$, $x=1\Rightarrow u=3$. $${\int }_0^1{x}^3\sqrt{1+2x^4}dx=\frac{1}{8}{\int }_1^3{u}^{1/2}du=\frac{1}{8}{\left[\frac{2}{3}{u}^{3/2}\right]}_1^3=\frac{1}{12}(3^{3/2}-1).$$

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In the Real World

1. eSewa: Uses definite integrals with substitution to calculate cumulative transaction fees for bulk payments. For example, if the fee for a transaction of size $x$ is $0.01x^2$, the total fee from $x=1$ to $x=1000$ is computed via ${\int }_1^{1000}0.01x^{2}dx$.
2. Pathao: Applies integration by parts to model driver waiting times. If $W(t)=t\ln t$ represents waiting time at time $t$, Pathao computes ${\int }_0^{T}W(t)dt$ to optimize delivery schedules.
3. NTC: Employs partial fractions to decompose rational functions modeling network delays. For instance, $\frac{2x+1}{x^2-x}$ is split into $\frac{1}{x}+\frac{1}{x-1}$ to analyze packet delays analytically.

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Exam Tip

1. Substitution: Always show the substitution step ($u=\dots$, $du=\dots$) and the reversed limits if definite.
2. Integration by Parts: Write the LIATE choice for $u$ and $dv$ explicitly. If the integral repeats, use the tabular method (e.g., for $x^2{e}^x$).
3. Partial Fractions: Factor the denominator completely and label each constant (e.g., $A$, $B$) clearly. Solve for constants by plugging roots or comparing coefficients.
4. Trigonometric Integrals: Rewrite powers using identities (e.g., ${\sin }^{2}x=\frac{1-\cos 2x}{2}$) before substituting.
5. Area Problems: For circles/ellipses, use symmetry to reduce the integral (e.g., $4\times {\int }_0^{a}f(x)dx$).
6. Units: In real-world problems (e.g., NTC delays), ensure the integral’s result has the correct units (e.g., seconds for time).

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Key Formula Summary: