12. Techniques of Integration

Verify the integration formulas in Exercises 111–114.

111. ∫ (arctan x) / x² dx = ln x - 1/2 ln(1 + x²) - arctan x / x + C

75. {Use of Tech} Oscillator displacements Suppose a mass on a spring that is slowed by friction has the position function:

s(t) = e⁻ᵗ sin t

c. Generalize part (b) and find the average value of the position on the interval [nπ, (n+1)π], for n = 0, 1, 2, ...

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ 4x sec²(2x) dx

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫(from 1 to e) x³ ln(x) dx

Evaluate the integrals in Exercises 1–8 using integration by parts.

∫ arccos(x / 2) dx

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ t² e^(4t) dt

92–98. Evaluate the following integrals.

92. ∫[1 to √2] y⁸ e^(y²) dy

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

∫ θ·cos(2θ + 1) dθ

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

∫ cotx·csc³x dx

78. Practice with tabular integration Evaluate the following integrals using tabular integration (refer to Exercise 77).

e. ∫ (2x² - 3x) / (x - 1)³ dx

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ x² sin(x) dx

Evaluate the integrals in Exercises 1–8 using integration by parts.

∫ x² ln(x) dx

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫₀¹/√2 2x arcsin(x²) dx

19. Center of mass Find the center of mass of a thin plate of constant density covering the region in the first and fourth quadrants enclosed by the curves y=1/(1+x²) and y=-1/(1+x²) and by the lines x=0 and x=1.

9–40. Integration by parts Evaluate the following integrals using integration by parts.

29. ∫ e⁻ˣ sin(4x) dx