Calculus: Trigonometric Integrals and Substitution Method

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Questions and Answers

Evaluate the integral ∫sin^3(x)dx using the reduction formula.

∫sin^3(x)dx = (-1)^3 ∫cos^3(x)dx = -∫cos^3(x)dx = -(1/3)cos^2(x)sin(x) + (2/3)∫cos(x)dx = -(1/3)cos^2(x)sin(x) + (2/3)sin(x) + C

Use the substitution method to evaluate the integral ∫(x^2 + 1)e^x dx.

Let u = x^2 + 1, then du/dx = 2x + 1. Substituting, we get ∫(x^2 + 1)e^x dx = ∫ue^u du = e^u + C = e^(x^2 + 1) + C

Approximate the definite integral ∫[0,1] e^x dx using the trapezoidal rule with n = 4.

Divide the interval [0,1] into 4 subintervals. Then, approximate the area using the trapezoidal rule: ∫[0,1] e^x dx ≈ (1/2)h(y0 + 2y1 + 2y2 + 2y3 + y4) = (1/2)(1/4)(e^0 + 2e^(1/4) + 2e^(2/4) + 2e^(3/4) + e^1) ≈ 1.718

Evaluate the integral ∫x^2 cos(x)dx using integration by parts.

<p>Let u = x^2, then dv = cos(x)dx. Then, du/dx = 2x, and v = sin(x). Using the integration by parts formula, we get ∫x^2 cos(x)dx = x^2 sin(x) - 2∫x sin(x)dx = x^2 sin(x) - 2(-x cos(x) + ∫cos(x)dx) = x^2 sin(x) + 2x cos(x) - 2sin(x) + C</p> Signup and view all the answers

Determine the convergence of the improper integral ∫[0,∞) e^(-x^2) dx.

<p>The integral converges by the p-test, since ∫[0,∞) e^(-x^2) dx &lt; ∫[0,∞) e^(-x) dx = 1/(e^(-0) - 0) = 1.</p> Signup and view all the answers

Evaluate the integral ∫cos^4(x)dx using the power reduction formula.

<p>∫cos^4(x)dx = (1/4)cos^3(x)sin(x) + (3/4)∫cos^2(x)dx = (1/4)cos^3(x)sin(x) + (3/4)((1/2)x + (1/2)sin(2x)) + C</p> Signup and view all the answers

Use the substitution method to evaluate the integral ∫(2x+1)/(x^2 + 1) dx.

<p>Let u = x^2 + 1, then du/dx = 2x + 1. Substituting, we get ∫(2x+1)/(x^2 + 1) dx = ∫1/u du = ln|u| + C = ln|x^2 + 1| + C</p> Signup and view all the answers

Approximate the definite integral ∫[0,π] sin(x)dx using Simpson's rule with n = 4.

<p>Divide the interval [0,π] into 4 subintervals. Then, approximate the area using Simpson's rule: ∫[0,π] sin(x)dx ≈ (1/3)h(y0 + 4y1 + 2y2 + 4y3 + y4) = (1/3)(π/4)(sin(0) + 4sin(π/4) + 2sin(π/2) + 4sin(3π/4) + sin(π)) ≈ 2</p> Signup and view all the answers

Evaluate the integral ∫x^3 sin(x)dx using integration by parts.

<p>Let u = x^3, then dv = sin(x)dx. Then, du/dx = 3x^2, and v = -cos(x). Using the integration by parts formula, we get ∫x^3 sin(x)dx = -x^3 cos(x) + 3∫x^2 cos(x)dx = -x^3 cos(x) + 3(x^2 sin(x) - 2∫x sin(x)dx) = -x^3 cos(x) + 3x^2 sin(x) - 6x cos(x) + 6sin(x) + C</p> Signup and view all the answers

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Study Notes

Trigonometric Integrals

  • Integrals involving trigonometric functions:
    • ∫sin(x)dx = -cos(x) + C
    • ∫cos(x)dx = sin(x) + C
    • ∫sin^2(x)dx = (1/2)x - (1/2)sin(2x) + C
    • ∫cos^2(x)dx = (1/2)x + (1/2)sin(2x) + C
  • Reduction formulas for trigonometric integrals:
    • ∫sin^n(x)dx = (-1)^n ∫cos^n(x)dx
    • ∫cos^n(x)dx = (1/n)cos^(n-1)(x)sin(x) + (n-1)/n ∫cos^(n-2)(x)dx

Substitution Method

  • Substitution method for integration:
    • Substitute u = f(x) and du/dx = f'(x) to transform the integral
    • ∫f(x)dx = ∫u(du/dx)dx = ∫udu
    • Example: ∫(2x+1)dx = ∫u(du/dx)dx = ∫udu = (1/2)u^2 + C

Numerical Integration

  • Approximating definite integrals using numerical methods:
    • Rectangular rule: approximate area by summing rectangular areas
    • Trapezoidal rule: approximate area by summing trapezoidal areas
    • Simpson's rule: approximate area by summing parabolic areas
    • Monte Carlo methods: approximate area by generating random points

Integration By Parts

  • Integration by parts formula:
    • ∫udv = uv - ∫vdu
    • Choose u and dv such that the integral becomes easier to evaluate
    • Example: ∫x^2 sin(x)dx = -x^2 cos(x) + 2∫x cos(x)dx

Improper Integrals

  • Improper integrals: integrals with infinite limits or integrands:
    • Types of improper integrals:
      • Infinite limits: ∫[a,∞) or ∫(-∞,b]
      • Integrands with infinite discontinuities: ∫[a,b] f(x)dx where f(x) has infinite discontinuities
    • Convergence tests:
      • Direct comparison test
      • Limit comparison test
      • P-test

Trigonometric Integrals

  • Basic trigonometric integrals:
    • ∫sin(x)dx = -cos(x) + C
    • ∫cos(x)dx = sin(x) + C
    • ∫sin^2(x)dx and ∫cos^2(x)dx can be evaluated using power reduction formulas
  • Power reduction formulas for trigonometric integrals:
    • ∫sin^n(x)dx = (-1)^n ∫cos^n(x)dx
    • ∫cos^n(x)dx = (1/n)cos^(n-1)(x)sin(x) + (n-1)/n ∫cos^(n-2)(x)dx

Substitution Method

  • Substitution method involves:
    • Substituting u = f(x) and du/dx = f'(x) to transform the integral
    • Rewriting the integral in terms of u and du
    • Example: ∫(2x+1)dx = ∫u(du/dx)dx = ∫udu = (1/2)u^2 + C

Numerical Integration

  • Approximating definite integrals using numerical methods:
    • Rectangular rule: approximate area by summing rectangular areas
    • Trapezoidal rule: approximate area by summing trapezoidal areas
    • Simpson's rule: approximate area by summing parabolic areas
    • Monte Carlo methods: approximate area by generating random points

Integration By Parts

  • Integration by parts formula:
    • ∫udv = uv - ∫vdu
    • Choosing u and dv such that the integral becomes easier to evaluate
    • Example: ∫x^2 sin(x)dx = -x^2 cos(x) + 2∫x cos(x)dx

Improper Integrals

  • Types of improper integrals:
    • Infinite limits: ∫[a,∞) or ∫(-∞,b]
    • Integrands with infinite discontinuities: ∫[a,b] f(x)dx where f(x) has infinite discontinuities
  • Convergence tests for improper integrals:
    • Direct comparison test
    • Limit comparison test
    • P-test

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