11. Integrals of Inverse, Exponential, & Logarithmic Functions

Evaluate the integrals in Exercises 31–78.

58. ∫(from 0 to ln9)e^θ(e^θ-1)^(1/2) dθ

Find the area under the graph of $f\left(x\right)=3^{2x}$ between $x=0$ and $x=2$.

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.)

7–84. Evaluate the following integrals.

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

Evaluate the indefinite integral.

${\int }_{}^{}3x^4-{\left(5\right)}^{x}dx$

"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."

Evaluate the definite integral.

${\int }_0^5\frac{3^{\sqrt{x+4}}}{\sqrt{x+4}}dx$

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 the integrals in Exercises 87–96.

93. ∫₀^(π/2) 7^(cos t) sin t dt

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

∫ 3^{-2x} dx