Chapter 6

Verify each inequality without evaluating the integrals. $$ \int_{0}^{1} x d x \geq \int_{0}^{1} x^{2} d x $$

Compute the indefinite integrals. $$ \int\left(1-\frac{x^{2}}{1+x^{2}}\right) d x $$

Verify each inequality without evaluating the integrals. $$ \int_{1}^{2} x d x \leq \int_{1}^{2} x^{2} d x $$

Compute the indefinite integrals. $$ \int \frac{1}{\sqrt{1-x^{2}}} d x $$

Verify each inequality without evaluating the integrals. $$ 0 \leq \int_{0}^{4} \sqrt{x} d x \leq 8 $$

Compute the indefinite integrals. $$ \int \frac{5}{\sqrt{1-x^{2}}} d x $$

Verify each inequality without evaluating the integrals. $$ \frac{1}{2} \leq \int_{0}^{1} \sqrt{1-x^{2}} d x \leq 1 $$

Compute the indefinite integrals. $$ \int \frac{1}{x+2} d x $$

Verify each inequality without evaluating the integrals. $$ \frac{\pi}{3} \leq \int_{\pi / 6}^{5 \pi / 6} \sin x d x \leq \frac{2 \pi}{3} $$

Compute the indefinite integrals. $$ \int \frac{1}{x-3} d x $$

Find the value of \(a \geq 0\) that maximizes \(\int_{0}^{a}\left(4-x^{2}\right) d x\).

Compute the indefinite integrals. $$ \int \frac{2 x-1}{3 x} d x $$

Find the value of \(a \in[0,2 \pi]\) that maximizes \(\int_{0}^{a} \cos x d x\).

Compute the indefinite integrals. $$ \int \frac{2 x+5}{x} d x $$

Find \(a \in(0,2 \pi]\) such that $$ \int_{0}^{a} \sin x d x=0 $$

Compute the indefinite integrals. $$ \int \frac{x+3}{x^{2}-9} d x $$

Find \(a>1\) such that $$ \int_{1}^{a}(x-2)^{3} d x=0 $$

Compute the indefinite integrals. $$ \int \frac{x+4}{x^{2}-16} d x $$

Find \(a>0\) such that $$ \int_{-a}^{a}(1-|x|) d x=0 $$

Compute the indefinite integrals. $$ \int \frac{3-x}{x^{2}-9} d x $$

To determine age-specific mortality, a group of individuals, all born at the same time, is followed over time. If \(N(t)\) denotes the number still alive at time \(t\), then \(N(t) / N(0)\) is the fraction surviving at time \(t\). The quantity \(r(t)\), called the hazard rate function, measures the rate at which individuals die at time \(t\); that is, \(r(t) d t\) is the probability that an individual who is alive at time \(t\) dies during the infinitesimal time interval \((t, t+d t)\). The cumulative hazard during the time interval \([0, t], \int_{0}^{t} r(s) d s\), can be estimated as \(-\ln \frac{N(t)}{N(0)}\). Show that the cumulative hazard during the time interval \([t, t+1], \int_{t}^{t+1} r(s) d s\), can be estimated as \(-\ln \frac{N(t+1)}{N(t)}\).

Compute the indefinite integrals. $$ \int \frac{4-x}{x^{2}-16} d x $$

Compute the indefinite integrals. $$ \int \frac{5 x^{2}}{x^{2}+1} d x $$

Compute the indefinite integrals. $$ \int \frac{2 x^{2}}{1+x^{2}} d x $$

Compute the indefinite integrals. $$ \int 3^{x} d x $$

Compute the indefinite integrals. $$ \int 2^{x} d x $$

Compute the indefinite integrals. $$ \int 3^{-2 x} d x $$

Compute the indefinite integrals. $$ \int 4^{-x} d x $$

Compute the indefinite integrals. $$ \int\left(x^{2}+2^{x}\right) d x $$

Compute the indefinite integrals. $$ \int\left(x^{-3}+3^{-x}\right) d x $$

Compute the indefinite integrals. $$ \int\left(\sqrt{x}+\sqrt{e^{x}}\right) d x $$

Compute the indefinite integrals. $$ \int\left(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{e^{x}}}\right) d x $$

Evaluate the definite integrals. $$ \int_{2}^{4}(3-2 x) d x $$

Evaluate the definite integrals. $$ \int_{-1}^{3}\left(2 x^{2}-1\right) d x $$

Evaluate the definite integrals. $$ \int_{0}^{1}\left(x^{3}-x^{1 / 3}\right) d x $$

Evaluate the definite integrals. $$ \int_{1}^{2} x^{5 / 2} d x $$

Evaluate the definite integrals. $$ \int_{1}^{8} x^{-2 / 3} d x $$

Evaluate the definite integrals. $$ \int_{4}^{9} \frac{1+\sqrt{x}}{\sqrt{x}} d x $$

Evaluate the definite integrals. $$ \int_{0}^{2}(2 t-1)(t+3) d t $$

Evaluate the definite integrals. $$ \int_{-1}^{2}(2+3 t)^{2} d t $$

Evaluate the definite integrals. $$ \int_{0}^{\pi / 4} \sin (2 x) d x $$

Evaluate the definite integrals. $$ \int_{-\pi / 3}^{\pi / 3} 2 \cos \left(\frac{x}{2}\right) d x $$

Evaluate the definite integrals. $$ \int_{0}^{\pi / 8} \sec ^{2}(2 x) d x $$

Evaluate the definite integrals. $$ \int_{-\pi / 4}^{\pi / 4} \tan x d x $$

Evaluate the definite integrals. $$ \int_{0}^{1} \frac{1}{1+x^{2}} d x $$

Evaluate the definite integrals. $$ \int_{-\sqrt{3}}^{-1} \frac{4}{1+x^{2}} d x $$

Evaluate the definite integrals. $$ \int_{0}^{1 / 2} \frac{1}{\sqrt{1-x^{2}}} d x $$

Evaluate the definite integrals. $$ \int_{-1 / 2}^{1 / 2} \frac{2}{\sqrt{1-x^{2}}} d x $$

Evaluate the definite integrals. $$ \int_{0}^{\pi / 6} \tan (2 x) d x $$

Evaluate the definite integrals. $$ \int_{\pi / 20}^{\pi / 15} \sec (5 x) \tan (5 x) d x $$