11. Integrals of Inverse, Exponential, & Logarithmic Functions

Find the area of $f\left(x\right)=3+e^{2x}$ from $x=0$ to $x=2$.

7–84. Evaluate the following integrals.

33. ∫ [eˣ / (a² + e²ˣ)] dx, where a ≠ 0

Evaluate the following integrals.

∫ e³ˣ/(eˣ - 1) dx

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

40. ∫ (e³ᵗ / √(4 + e²ᵗ)) dt

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?