11. Integrals of Inverse, Exponential, & Logarithmic Functions

Evaluate ∫ 4ˣ dx.

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.

∫ 3^{-2x} dx

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.

∫₀ˡⁿ ² (e^{3x} − e^{−3x}) / (e^{3x} + e^{−3x}) dx

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.

∫₀^{π/2} 4^{sin x} cos x dx

37–56. Integrals Evaluate each integral.

∫ eˣ/(36 – e²ˣ), x < ln 6

Energy consumption On the first day of the year (t=0), a city uses electricity at a rate of 2000 MW. That rate is projected to increase at a rate of 1.3% per year.

b. Find the total energy (in MW-yr) used by the city over four full years beginning at t=0.

Energy consumption On the first day of the year (t=0), a city uses electricity at a rate of 2000 MW. That rate is projected to increase at a rate of 1.3% per year.

c. Find a function that gives the total energy used (in MW-yr) between t=0 and any future time t>0.

2–9. Integrals Evaluate the following integrals.

∫ (eˣ / (4eˣ + 6)) dx

2–9. Integrals Evaluate the following integrals.

∫₁⁴ (10^{√x} / √x) dx

Oil consumption Starting in 2018 (t=0), the rate at which oil is consumed by a small country increases at a rate of 1.5%/yr, starting with an initial rate of 1.2 million barrels/yr.

b. Find the function that gives the amount of oil consumed between t=0 and any future time t.

Oil consumption Starting in 2018 (t=0), the rate at which oil is consumed by a small country increases at a rate of 1.5%/yr, starting with an initial rate of 1.2 million barrels/yr.

c. How many years after 2018 will the amount of oil consumed since 2018 reach 10 million barrels?

Many formulas There are several ways to express the indefinite integral of sech x.

b. Show that ∫ sech x dx = sin⁻¹ (tanh x) + C. (Hint: Show that sech x = sech² x / √(1 − tanh² x) and then make a change of variables.)

"Integral formula Carry out the following steps to derive the formula ∫ csch x dx = ln |tanh(x / 2)| + C (Theorem 7.6).

b. Use the identity for sinh(2u) to show that 2 / sinh(2u) = sech² u / tanh u."