11. Integrals of Inverse, Exponential, & Logarithmic Functions

Find the indefinite integral.

${\int }_{}^{}\frac{3-y^2}{2y}dy$

Evaluate the definite integral.

${\int }_2^e\frac{3+x}{x}dx$

Find the area under the graph of $f\left(t\right)=\frac{t^2{e}^t+t}{t^2}$ from $t=1$ to $t=3$.

Find the indefinite integral.

${\int }_{}^{}\frac{1}{2x+5}dx$

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

19. ∫ 1/√(x² - 81) dx, x > 9

Clever substitution Evaluate ∫ dx/(1 + sin x + cos x) using the substitution x=2 tan⁻¹ θ. The identities sin x = 2 sin(x/2) cos(x/2) and cos x =cos²(x/2) − sin²(x/2) are helpful.

Zero net area Consider the function f(x) = (1 − x)/x

a. Are there numbers 0 < a < 1 such that ∫₁₋ₐ¹⁺ᵃ f(x) dx = 0?

37–56. Integrals Evaluate each integral.

∫ dx/(8 – x²), x > 2√2