12. Techniques of Integration

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

b. ∫ 7x e³ˣ dx

82. A family of exponentials The curves y = x * e^(-a * x) are shown in the figure for a = 1, 2, and 3.

e. Does this pattern continue? Is it true that A(1, ln b) = a² * A(a, (ln b)/a)?

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, ...

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.

∫ √(x - 2) / √(x - 1) dx

Use any method to evaluate the integrals in Exercises 65–70.

∫ x cos³(x) dx

Evaluate the limits in Exercise 9 and 10 by identifying them with definite integrals and evaluating the integrals.

lim (n → ∞) Σ (from k=1 to n) ln √(1 + k/n)

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

∫ x² sin(x) dx

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

∫ (x² - 2x + 1) e^(2x) dx

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

∫ arcsin(y) dy

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.

∫ (r² + r + 1) e^r dr

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

∫ t² e^(4t) dt

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

∫ x⁵ e³ˣ dx

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

∫ √x e√x dx

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

∫₀^π/2 x³ cos 2x dx