7. Antiderivatives & Indefinite Integrals

9–61. Trigonometric integrals Evaluate the following integrals.

51. ∫ (csc²x + csc⁴x) dx

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

 ∫ sec 4w tan 4w dw

9–61. Trigonometric integrals Evaluate the following integrals.

34. ∫ tan⁹x sec⁴x dx

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

133. ∫ (sin²x) / (1 + sin²x) dx

Evaluate the integral: ${\int }_1^7{w}^2\ln \left(w\right) dw$

Evaluate the integrals in Exercises 39–56.

56. ∫sec(x)dx/√(ln(sec(x)+tan(x)))

7–84. Evaluate the following integrals.

53. ∫ eˣ cot³(eˣ) dx

Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

∫ dx / (1 + sin x + cos x)

Evaluate the integrals in Exercises 33–54.

∫₀^(π/4) (1 + e^(tan θ)) sec²θ dθ

Evaluate the integrals in Exercises 1–22.

∫ 7cos⁷(t) dt

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ (sin5t) dt / [1 + (cos5t)²]

Evaluate the double integral by reversing the order of integration: ${\int }_0^4{\int }_{3y}^{12}13e^{x^2} dx dy$

7–64. Integration review Evaluate the following integrals.

53. ∫ eˣ sec(eˣ + 1) dx

Evaluate the integral: ${\int }_0^{\frac{\pi }{2}}3{\cos }^2\left(x\right) dx$.

Evaluate the integrals in Exercises 31–78.

47. ∫(1/r)csc²(1+ln(r))dr