11. Integrals of Inverse, Exponential, & Logarithmic Functions

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.

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

Evaluate the definite integral.

${\int }_1^6\frac{dt}{2+5t}$

Evaluate the definite integral.

${\int }_0^{\frac{\pi }{3}}\frac{sin\theta }{1+\cos \theta }d\theta$

Find the area under the graph of $f\left(x\right)=\frac{e^{-x}}{1+e^{-x}}$ from $x=0$ to $x=4$.

The region between the curve $y=\frac{2}{\sqrt{x}}$ and the $x$-axis from $x=1$ to $x=3$ is revolved about the $x$-axis to form a solid. Find the volume of this solid. 

Solve the initial value problem given by $\frac{dy}{dx}=\frac{2}{x}+3,y\left(1\right)=4$.

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

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

∫₁² (1 + ln x) x^x dx

63–68. Definite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms.

∫₁ᵉ^² dx/x√(ln²x + 1)

2–9. Integrals Evaluate the following integrals.

∫ (x + 4) / (x² + 8x + 25) dx

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.

e. The area under the curve y = 1/x and the x-axis on the interval [1, e] is 1.