The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (2^(√y) dy) / 2√y

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."

Evaluate the integrals in Exercises 87–96.

87. ∫ 5ˣ dx

Evaluate the integrals in Exercises 31–78.

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

Evaluate the integrals in Exercises 87–96.

89. ∫₀¹ 2^(−θ) dθ

Evaluate the integrals in Exercises 87–96.

91. ∫₁^(√2) x·2^(x²) dx

Evaluate the integrals in Exercises 87–96.

93. ∫₀^(π/2) 7^(cos t) sin t dt

Evaluate the integrals in Exercises 31–78.

49. ∫x3^(x²)dx