Integration of Algebraic, Exponential, and Trigonometric Functions
FORMULAS USED:
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Power Rule of Linear Function:
\int (ax + b)^n \, dx = \frac{(ax + b)^{n+1}}{a(n+1)} + C \quad (n \neq -1)
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Logarithmic Rule:
\int \frac{1}{ax + b} \, dx = \frac{1}{a} \ln|ax + b| + C
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Exponential Rule:
\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C
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Trigonometric Rules:
- \int \sin(ax + b) \, dx = -\frac{1}{a}\cos(ax + b) + C
- \int \cos(ax + b) \, dx = \frac{1}{a}\sin(ax + b) + C
- \int \sec(ax)\tan(ax) \, dx = \frac{1}{a}\sec(ax) + C
- \int \csc^2(ax + b) \, dx = -\frac{1}{a}\cot(ax + b) + C
SOLUTION STEPS:
Section 1: Algebraic Integrals
Step 1 — Solve 1.(i) \int (1-2x)^4 \, dx
Step 2 — Solve 1.(ii) \int (4+5x)^3 \, dx
Step 3 — Solve 1.(iii) \int \frac{1}{(1+x)^4} \, dx
Step 4 — Solve 1.(iv) \int \sqrt{ax+b} \, dx
Step 5 — Solve 1.(v) \int \frac{1}{2x+3} \, dx
Step 6 — Solve 1.(vi) \int (1-4x)^{-12} \, dx
Step 7 — Solve 1.(vii) \int \frac{1}{\sqrt{1+x}} \, dx
Step 8 — Solve 1.(viii) \int \frac{1}{3-5x} \, dx
Step 9 — Solve 1.(ix) \int \sqrt{1+x} \, dx
Step 10 — Solve 1.(x) \int \frac{1}{(1-3x)^2} \, dx
Section 2: Exponential Integrals
Step 11 — Solve 2.(i) \int e^{mx} \, dx
Step 12 — Solve 2.(ii) \int e^{-x} \, dx
Step 13 — Solve 2.(iii) \int e^{3x} \, dx
Section 3: Trigonometric Integrals
Step 14 — Solve 3.(i) \int \sin(mx) \, dx
Step 15 — Solve 3.(ii) \int \sin\left(\frac{x}{2}\right) \, dx
Step 16 — Solve 3.(iii) \int \sin(3x) \, dx
Step 17 — Solve 3.(iv) \int \sin(2x+3) \, dx
Step 18 — Solve 3.(v) \int \cos(2x-3) \, dx
Step 19 — Solve 3.(vi) \int \cos\left(\frac{x}{2}\right) \, dx
Step 20 — Solve 3.(vii) \int \cos(2x) \, dx
Step 21 — Solve 3.(viii) \int \sec(3x)\tan(3x) \, dx
Step 22 — Solve 3.(ix) \int \cos\left(\frac{\pi}{6}+2x\right) \, dx
Step 23 — Solve 3.(x) \int \csc^2(1-3x) \, dx
Step 24 — Solve 3.(xi) \int \sin\left(2x+\frac{\pi}{4}\right) \, dx
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FINAL ANSWERS SUMMARY:
- 1.(i): -\frac{1}{10}(1-2x)^5 + C
- 1.(ii): \frac{1}{20}(4+5x)^4 + C
- 1.(iii): -\frac{1}{3(1+x)^3} + C
- 1.(iv): \frac{2}{3a}(ax+b)^{3/2} + C
- 1.(v): \frac{1}{2}\ln|2x+3| + C
- 1.(vi): \frac{1}{44(1-4x)^{11}} + C
- 1.(vii): 2\sqrt{1+x} + C
- 1.(viii): -\frac{1}{5}\ln|3-5x| + C
- 1.(ix): \frac{2}{3}(1+x)^{3/2} + C
- 1.(x): \frac{1}{3(1-3x)} + C
- 2.(i): \frac{1}{m}e^{mx} + C
- 2.(ii): -e^{-x} + C
- 2.(iii): \frac{1}{3}e^{3x} + C
- 3.(i): -\frac{1}{m}\cos(mx) + C
- 3.(ii): -2\cos\left(\frac{x}{2}\right) + C
- 3.(iii): -\frac{1}{3}\cos(3x) + C
- 3.(iv): -\frac{1}{2}\cos(2x+3) + C
- 3.(v): \frac{1}{2}\sin(2x-3) + C
- 3.(vi): 2\sin\left(\frac{x}{2}\right) + C
- 3.(vii): \frac{1}{2}\sin(2x) + C
- 3.(viii): \frac{1}{3}\sec(3x) + C
- 3.(ix): \frac{1}{2}\sin\left(2x+\frac{\pi}{6}\right) + C
- 3.(x): \frac{1}{3}\cot(1-3x) + C
- 3.(xi): -\frac{1}{2}\cos\left(2x+\frac{\pi}{4}\right) + C
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KEY CONCEPTS:
1. Linear Substitution in Integration
- Definition: When an integrand contains a linear term (ax+b) instead of x, the integral of f(ax+b) is \frac{1}{a}F(ax+b) + C, where F'(x) = f(x).
- In this problem: We divided each standard anti-derivative by the coefficient of x (the value of a).
COMMON MISTAKES:
Forgetting to divide by the coefficient of x
- Wrong: \int \sin(3x) \, dx = -\cos(3x) + C
- Right: \int \sin(3x) \, dx = -\frac{1}{3}\cos(3x) + C
- Why it’s wrong: Differentiating -\cos(3x) by the chain rule gives 3\sin(3x), so the factor of \frac{1}{3} is required to cancel the 3.
Sign errors with negative x coefficients
- Wrong: \int \frac{1}{3-5x} \, dx = \frac{1}{5} \ln|3-5x| + C
- Right: \int \frac{1}{3-5x} \, dx = -\frac{1}{5} \ln|3-5x| + C
- Why it’s wrong: The coefficient of x is -5, so the factor in front must be -\frac{1}{5}.
Would you like me to walk you through a step-by-step u-substitution proof for any specific problem from this set?
