Calculus II: Chapter 7 Integration Techniques Review

Chapter 7 Review: Comprehensive Integration Techniques

This review covers a wide array of integration techniques, including integration by parts, trigonometric integrals, substitution methods, hyperbolic functions, partial fractions, trigonometric substitution, definite integrals, improper integrals, and numerical approximation.

Indefinite Integrals: Integration by Parts and Related Techniques

1. Problem Statement Unclear; Result Provided as:
   5ye - 25e + C

2. Integration of x imes ext{trigonometric function}:

   • Integral: `\int x \sin(\pi x) dx

   • Solution: -\frac{1}{\pi}x \cos(\pi x) + \frac{1}{\pi^2} \sin(\pi x) + C

     • (Note: The provided solution (x-1) \cos(\pi x) + \frac{1}{\pi} \sin(\pi x) + C seems to be an error in transcription or calculation, as standard integration by parts \int u dv = uv - \int v du with u=x, dv=\sin(\pi x) dx leads to the corrected solution above.)

3. Integral of \cos^3 x:

   • Integral: `\int \cos^3 x dx

   • Given Solution: x \cos^3 x - (1 - \frac{1}{2}x^2) + C

     • (Note: This solution does not align with standard techniques for \int \cos^3 x dx, which typically involves \sin x - \frac{1}{3} \sin^3 x + C.)

4. Integral of \sin(\sqrt{x}) (Substitution and Integration by Parts):

   • Integral: `\int \sin(\sqrt{x}) dx

   • Solution: 2\sqrt{x}\sin(\sqrt{x}) + 2\cos(\sqrt{x}) + C

     • (Method: Let u = \sqrt{x}, then x=u^2 and dx = 2u du. The integral becomes 2\int u \sin u du, which is solved using integration by parts.)

5. Integral of t e^{nt}:

   • Integral: `\int t e^{nt} dt

   • Solution: \frac{1}{n} t e^{nt} - \frac{1}{n^2} e^{nt} + C

     • (Method: Integration by parts with u=t, dv=e^{nt} dt.)

6. Integral of (x^2+2x) \cos x (Repeated Integration by Parts):

   • Integral: `\int (x^2+2x) \cos x dx

   • Solution: \left(x^2+2x\right) \sin x + (2x+2) \cos x - 2 \sin x + C

     • (Method: Requires two applications of integration by parts, differentiating x^2+2x and integrating \cos x.)

7. Integral of t^2 \sin(\beta t) (Repeated Integration by Parts):

   • Integral: `\int t^2 \sin(\beta t) dt

   • Solution: -\frac{t^2}{\beta} \cos(\beta t) + \frac{2t}{\beta^2} \sin(\beta t) + \frac{2}{\beta^3} \cos(\beta t) + C

8. Integral of x \cosh(ax) (Integration by Parts with Hyperbolic Functions):

   • Integral: `\int x \cosh(ax) dx

   • Solution: \frac{x}{a} \sinh(ax) - \frac{1}{a^2} \cosh(ax) + C

9. Integral of y \sinh(2y) (Integration by Parts with Hyperbolic Functions):

   • Integral: `\int y \sinh(2y) dy

   • Solution: \frac{y}{2} \cosh(2y) - \frac{1}{4} \sinh(2y) + C

10. Integral of x e^{2x} (Integration by Parts):

    • Integral: `\int x e^{2x} dx

    • Solution: \frac{e^{2x}}{4}(2x-1) + C (or \frac{1}{2}x e^{2x} - \frac{1}{4}e^{2x} + C)

11. Integral of \frac{e^{\sqrt{x}}}{\sqrt{x}} (Substitution):

    • Integral: `\int \frac{e^{\sqrt{x}}}{\sqrt{x}} dx

    • Solution: 2e^{\sqrt{x}} + C

      • (Method: Let u = \sqrt{x}, then du = \frac{1}{2\sqrt{x}} dx.)

12. Integral of x^2 \ln x (Integration by Parts):

    • Integral: `\int x^2 \ln x dx

    • Solution: \frac{x^3}{3} \ln|x| - \frac{x^3}{9} + C

Indefinite Integrals: Trigonometric Powers and Products

1. Integral of \tan^2 \theta \sec^4 \theta (Trigonometric Power):

   • Integral: `\int \tan^2 \theta \sec^4 \theta d\theta

   • Solution: \frac{1}{5}\tan^5 \theta + \frac{1}{3}\tan^3 \theta + C

     • (Method: \sec^4 \theta = \sec^2 \theta \cdot \sec^2 \theta = (1+\tan^2 \theta) \sec^2 \theta. Let u=\tan \theta.)

2. Integral of \sin^3 \theta \cos^{1/2} \theta (Trigonometric Power):

   • Integral: `\int \sin^3 \theta \cos^{1/2} \theta d\theta

   • Solution: -\frac{2}{3} \cos^{3/2} \theta + \frac{2}{7} \cos^{7/2} \theta + C

     • (Method: \sin^3 \theta = \sin^2 \theta \sin \theta = (1-\cos^2 \theta) \sin \theta. Let u=\cos \theta.)

3. Integral of \sin^3(2t) \cos^2(2t) (Trigonometric Power):

   • Integral: `\int \sin^3(2t) \cos^2(2t) dt

   • Solution: -\frac{1}{6} \cos^3(2t) + \frac{1}{10} \cos^5(2t) + C

     • (Method: Similar to above, \sin^3(2t) = (1-\cos^2(2t)) \sin(2t). Let u=\cos(2t).)

4. Integral of \sin^3 x \cos^{1/2} x (Trigonometric Power):

   • Integral: `\int \sin^3 x \cos^{1/2} x dx

   • Solution: -\frac{2}{3} \cos^{3/2} x + \frac{2}{7} \cos^{7/2} x + C

     • (This appears to be a repeat of problem 15 with \theta replaced by x.)

5. Integral of \cos^3(2t) (Trigonometric Power):

   • Integral: `\int \cos^3(2t) dt

   • Solution: \frac{1}{2} \sin(2t) - \frac{1}{6} \sin^3(2t) + C

     • (Method: \cos^3(2t) = (1-\sin^2(2t)) \cos(2t). Let u=\sin(2t).)

6. Integral of \tan^4 x \sec^6 x (Trigonometric Power):

   • Integral: `\int \tan^4 x \sec^6 x dx

   • Solution: \frac{1}{9} \tan^9 x + \frac{2}{7} \tan^7 x + \frac{1}{5} \tan^5 x + C

     • (Method: \sec^6 x = \sec^4 x \sec^2 x = (1+\tan^2 x)^2 \sec^2 x. Let u=\tan x.)

Indefinite Integrals: Trigonometric Product-to-Sum Identities

1. Integral of \sin x \sin(2x):

   • Integral: `\int \sin x \sin(2x) dx

   • Solution: \frac{1}{2} \sin^2 x + C (This solution is simplified from using product-to-sum: \frac{1}{2t}\cos t - \frac{1}{6}\cos 3t + C for \sin x \sin(2x) = \frac{1}{2}(\cos(x-2x) - \cos(x+2x)) = \frac{1}{2}(\cos(-x) - \cos(3x)) = \frac{1}{2}(\cos x - \cos(3x)))

     • (Correction: \int \sin x \sin(2x) dx = \int \sin x (2\sin x \cos x) dx = 2 \int \sin^2 x \cos x dx. Let u=\sin x, du=\cos x dx. Then 2 \int u^2 du = \frac{2}{3} u^3 + C = \frac{2}{3} \sin^3 x + C. The provided \frac{1}{2}\sin^2 x + C is likely incorrect for \sin x \sin(2x).)

2. Integral of \cos(8x) \cos(5x):

   • Integral: `\int \cos(8x) \cos(5x) dx

   • Solution: \frac{1}{26} \sin(13x) + \frac{1}{6} \sin(3x) + C

     • (Method: Use product-to-sum identity: \cos A \cos B = \frac{1}{2}[\cos(A-B) + \cos(A+B)].)

3. Integral of \sin(2\theta) \sin(6\theta):

   • Integral: `\int \sin(2\theta) \sin(6\theta) d\theta

   • Solution: -\frac{1}{8} \sin(4\theta) + \frac{1}{16} \sin(8\theta) + C

     • (Method: Use product-to-sum identity: \sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)].)

Indefinite Integrals: Trigonometric Substitution

1. Integral of \frac{\sqrt{9-x^2}}{x}:

   • Integral: `\int \frac{\sqrt{9-x^2}}{x} dx

   • Solution: \sqrt{9-x^2} - 3 \ln\left|\frac{3+\sqrt{9-x^2}}{x}\right| + C (or \sqrt{9-x^2} - 3 \operatorname{arcsech}(x/3) + C)

     • (Method: Let x = 3 \sin \theta.)

2. Integral of \frac{\sqrt{9-x^2}}{x^2}:

   • Integral: `\int \frac{\sqrt{9-x^2}}{x^2} dx

   • Solution: -\frac{\sqrt{9-x^2}}{x} - \arcsin\left(\frac{x}{3}\right) + C

     • (Method: Let x = 3 \sin \theta.)

3. Integral of \frac{dx}{x^2 \sqrt{4-x^2}}:

   • Integral: `\int \frac{dx}{x^2 \sqrt{4-x^2}}

   • Solution: -\frac{\sqrt{4-x^2}}{4x} + C

     • (Method: Let x = 2 \sin \theta.)

4. Integral of \frac{dx}{(x^2+4)^{3/2}}:

   • Integral: `\int \frac{dx}{(x^2+4)^{3/2}}

   • Solution: \frac{x}{4\sqrt{x^2+4}} + C

     • (Method: Let x = 2 \tan \theta.)

5. Integral of \frac{x^2}{\sqrt{9-x^2}} dx:

   • Integral: `\int \frac{x^2}{\sqrt{9-x^2}} dx

   • Solution: \frac{9}{2} \arcsin\left(\frac{x}{3}\right) - \frac{x}{2} \sqrt{9-x^2} + C

     • (Method: Let x = 3 \sin \theta.)

6. Integral of \frac{\sqrt{x^2-1}}{x^4} dx:

   • Integral: `\int \frac{\sqrt{x^2-1}}{x^4} dx

   • Solution: \frac{1}{3} \frac{(x^2-1)^{3/2}}{x^3} + C

     • (Method: Let x = \sec \theta.)

7. Integral of \frac{\sqrt{x^2-9}}{x^3} dx:

   • Integral: `\int \frac{\sqrt{x^2-9}}{x^3} dx

   • Solution: \frac{1}{6} \operatorname{arcsec}\left(\frac{|x|}{3}\right) - \frac{\sqrt{x^2-9}}{2x^2} + C

     • (Method: Let x = 3 \sec \theta.)

Indefinite Integrals: Partial Fractions and Completing the Square

1. Integral of \frac{x+2}{x^2+2x+5} dx (Completing the Square):

   • Integral: `\int \frac{x+2}{x^2+2x+5} dx

   • Solution: \frac{1}{2} \ln|x^2+2x+5| + \frac{1}{2} \arctan\left(\frac{x+1}{2}\right) + C

     • (Method: Complete the square in the denominator x^2+2x+5 = (x+1)^2+4. Split the numerator into parts that match the derivative of the denominator and a constant for \arctan.)

2. Integral of \frac{x^2+2x-1}{x(2x-1)(x+2)} dx (Partial Fractions):

   • Integral: `\int \frac{x^2+2x-1}{x(2x-1)(x+2)} dx

   • Solution: \frac{1}{2} \ln|x| + \frac{1}{10} \ln|2x-1| - \frac{1}{5} \ln|x+2| + C

3. Integral of \frac{x^3-2x^2+4x+1}{x^2-2x+4} dx (Polynomial Long Division and Partial Fractions):

   • Integral: `\int \frac{x^3-2x^2+4x+1}{x^2-2x+4} dx

   • Solution: \frac{x^2}{2} + \ln|x^2-2x+4| + C

     • (Method: Perform polynomial long division first, then integrate the resulting polynomial and the remainder.)

4. Integral of \frac{2x^2-x+4}{x(x^2+4)} dx (Partial Fractions):

   • Integral: `\int \frac{2x^2-x+4}{x(x^2+4)} dx

   • Solution: \ln|x| + \frac{1}{2}\ln(x^2+4) - \frac{1}{4} \arctan\left(\frac{x}{2}\right) + C

5. Integral of \frac{x^4}{x-1} dx (Polynomial Long Division):

   • Integral: `\int \frac{x^4}{x-1} dx

   • Solution: \frac{x^4}{4} + \frac{x^3}{3} + \frac{x^2}{2} + x + \ln|x-1| + C

6. Integral of \frac{x-4}{x^2-3x+2} dx (Partial Fractions):

   • Integral: `\int \frac{x-4}{x^2-3x+2} dx

   • Solution: 3 \ln|x-1| - 2 \ln|x-2| + C

     • (Method: Factor the denominator x^2-3x+2 = (x-1)(x-2).)

7. Integral of \frac{1}{1-\sin x} dx (Conjugate Multiplication):

   • Integral: `\int \frac{1}{1-\sin x} dx

   • Solution: \tan x + \sec x + C

     • (Method: Multiply numerator and denominator by 1+\sin x to get \frac{1+\sin x}{\cos^2 x} and split into \sec^2 x + \tan x \sec x.)

Indefinite Integrals: Mixed Techniques

1. Integral of t \tan^2 t dt (Integration by Parts and Identities):

   • Integral: `\int t \tan^2 t dt

   • Solution: t \tan t - \frac{1}{2} t^2 - \ln|\cos t| + C

     • (Method: Use \tan^2 t = \sec^2 t - 1. Then apply integration by parts to t (\sec^2 t - 1).)

2. Integral of \frac{dx}{x^3 \sqrt{x^2-1}} (Trigonometric Substitution):

   • Integral: `\int \frac{dx}{x^3 \sqrt{x^2-1}}

   • Solution: \frac{\sqrt{x^2-1}}{2x^2} + \frac{1}{2} \operatorname{arcsec}|x| + C

Definite Integrals

1. Definite Integral \int_{0}^{\pi} t \cos^2 t dt:

   • Integral: `\int_{0}^{\pi} t \cos^2 t dt

   • Solution: \frac{\pi^2}{4}

     • (Method: Use half-angle identity \cos^2 t = \frac{1+\cos(2t)}{2}, then integrate by parts.)

2. Definite Integral \int_{0}^{\pi} y \sin^2 y dy:

   • Integral: `\int_{0}^{\pi} y \sin^2 y dy

   • Solution: \frac{\pi^2}{4}

     • (Method: Use half-angle identity \sin^2 y = \frac{1-\cos(2y)}{2}, then integrate by parts.)

Numerical Integration

1. Approximation Using Simpson's Rule:

   • Problem: Approximate \int_{0}^{2} \frac{1}{1+x^2} dx with n=10

   • Result: 1.10714

Improper Integrals

1. Integral of \int_{1}^{\infty} \frac{1}{x^3} dx:

   • Integral: `\int_{1}^{\infty} \frac{1}{x^3} dx

   • Result: = \frac{1}{2} (Converges)

2. Integral of \int_{2}^{\infty} \frac{1}{x-2^{3/2}} dx:

   • Integral: `\int_{2}^{\infty} \frac{1}{(x-2)^{3/2}} dx

   • Result: Diverges (\infty)

3. Integral of \int_{0}^{1} \frac{1}{\sqrt{t}} dt:

   • Integral: `\int_{0}^{1} \frac{1}{\sqrt{t}} dt

   • Result: Converges (2)

4. Integral of \int_{0}^{\infty} y^3 e^{-3y^2} dy:

   • Integral: `\int_{0}^{\infty} y^3 e^{-3y^2} dy

   • Result: Converges (\frac{1}{54})

     • (Method: Requires substitution u=3y^2 and then integration by parts.)

5. Integral of \int_{0}^{\pi} \frac{1}{\sin^2 \theta} d\theta:

   • Integral: `\int{0}^{\pi} \frac{1}{\sin^2 \theta} d\theta = \int{0}^{\pi} \csc^2 \theta d\theta

   • Result: Diverges

     • (Explanation: \csc^2 \theta has vertical asymptotes at \theta=0 and \theta=\pi, making it an improper integral. \int \csc^2 \theta d\theta = -\cot \theta. Evaluating -\cot \theta at 0 and \pi shows divergence.)