Chapter 5

Verify by differentiation that the formula is correct. \(\int x \cos x d x=x \sin x+\cos x+C\)

Given that $$\int_{0}^{1} 3 x \sqrt{x^{2}+4} d x=5 \sqrt{5}-8$$, what is $$\int_{1}^{0} 3 u \sqrt{u^{2}+4} d u ?$$

Evaluate the indefinite integral. \(\int \frac{x}{1+x^{4}} d x\)

\(37-38\) Find the general indefinite integral. Illustrate by graphing several members of the family on the same screen. \(\int\left(\cos x+\frac{1}{2} x\right) d x\)

Suppose that \(f(1)=2, f(4)=7, f^{\prime}(1)=5, f^{\prime}(4)=3,\) and \(f^{\prime \prime}\) is continuous. Find the value of \(\int_{1}^{4} x f^{\prime \prime}(x) d x\)

Write as a single integral in the form $$\int_{a}^{b} f(x) d x$$: $$\int_{-2}^{2} f(x) d x+\int_{2}^{5} f(x) d x-\int_{-2}^{-1} f(x) d x$$

Evaluate the definite integral. \(\int_{0}^{1} \cos (\pi t / 2) d t\)

Find the general indefinite integral. Illustrate by graphing several members of the family on the same screen. \(\int\left(e^{x}-2 x^{2}\right) d x\)

If \(f(0)=g(0)=0\) and \(f^{\prime \prime}\) and \(g^{\prime \prime}\) are continuous, show that $$\int_{0}^{a} f(x) g^{\prime \prime}(x) d x=f(a) g^{\prime}(a)-f^{\prime}(a) g(a)+\int_{0}^{a} f^{\prime \prime}(x) g(x) d x$$

If $$\int_{1}^{5} f(x) d x=12$$ and $$\int_{4}^{5} f(x) d x=3.6$$, find $$\int_{1}^{4} f(x) d x$$.

Evaluate the definite integral. \(\int_{0}^{1}(3 t-1)^{50} d t\)

Find the general indefinite integral. \(\int(1-t)\left(2+t^{2}\right) d t\)

If $$\int_{0}^{9} f(x) d x=37$$ and $$\int_{0}^{9} g(x) d x=16$$ , find $$\int_{0}^{9}[2 f(x)+3 g(x)] d x$$.

Evaluate the definite integral. \(\int_{0}^{1} \sqrt[3]{1+7 x} d x\)

Find the general indefinite integral. \(\int v\left(v^{2}+2\right)^{2} d v\)

Find $$\int_{0}^{5} f(x) d x$$ if $$f(x)=\left\\{\begin{array}{ll}{3} & {\text { for } x<3} \\ {x} & {\text { for } x \geqslant 3}\end{array}\right.$$

Evaluate the definite integral. \(\int_{0}^{\sqrt{\pi}} x \cos \left(x^{2}\right) d x\)

Find the general indefinite integral. \(\int\left(1+\tan ^{2} \alpha\right) d \alpha\)

Evaluate the definite integral. \(\int_{0}^{1} x^{2}\left(1+2 x^{3}\right)^{5} d x\)

Find the general indefinite integral. \(\int \sec t(\sec t+\tan t) d t\)

Evaluate the definite integral. \(\int_{1 / 6}^{1 / 2} \csc \pi t \cot \pi t d t\)

Find the general indefinite integral. \(\int \frac{\sin x}{1-\sin ^{2} x} d x\)

Evaluate the definite integral. \(\int_{1}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} d x\)

Find the general indefinite integral. \(\int \frac{\sin 2 x}{\sin x} d x\)

Suppose \(f\) has absolute minimum value \(m\) and absolute maximum value \(M .\) Between what two values must \(\int_{0}^{2} f(x) d x\) lie? Which property of integrals allows you to make your conclusion?

Evaluate the definite integral. \(\int_{0}^{\pi / 2} \cos x \sin (\sin x) d x\)

Measles pathogenesis The function $$f(t)=-t(t-21)(t+1)$$ has been used to model the measles virus concentration in an infected individual. The area under the graph of \(f\) represents the total amount of infection. We saw in Section 5.1 that at \(t=12\) days this total amount of infection reaction 5.1 threshold beyond which symptoms appear. Use the Evaluation Theorem to calculate this threshold value.

Evaluate the definite integral. \(\int_{-\pi / 4}^{\pi / 4}\left(x^{3}+x^{4} \tan x\right) d x\)

If \(V^{\prime}(t)\) is the rate at which water flows into a reservoir at time \(t,\) what does the integral $$\int_{t_{1}}^{t_{2}} V^{\prime}(t) d t$$ represent?

Evaluate the definite integral. \(\int_{-\pi / 2}^{\pi / 2} \frac{x^{2} \sin x}{1+x^{6}} d x\)

Evaluate the definite integral. \(\int_{1}^{2} x \sqrt{x-1} d x\)

Age-structured populations Suppose the number of individuals of age \(a\) is given by the function \(N(a)\) (number of individuals per age \(a\) ). What does the integral \(\int_{0}^{15} N(a) d a\) represent?

Express the limit as a definite integral. $$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^{n} \frac{1}{1+(i / n)^{2}}$$

Evaluate the definite integral. \(\int_{0}^{a} x \sqrt{a^{2}-x^{2}} d x\)

Sea urchins Integration is sometimes used when censusing a population. For example, suppose the density of sea urchins at different points \(x\) along a coastline is given by the function \(f(x)\) individuals per meter, where \(x\) is the distance (in meters) along the coast from the start of the species' range. What does the integral \(\int_{a}^{b} f(x) d x\) represent?

Evaluate the definite integral. \(\int_{0}^{1} \frac{e^{z}+1}{e^{z}+z} d z\)

Bacteria growth A bacteria colony increases in size at a rate of 4.0553\(e^{1.8 t}\) bacteria per hour. If the initial population is 46 bacteria, find the population four hours later.

Evaluate the definite integral. \(\int_{0}^{T / 2} \sin (2 \pi t / T-\alpha) d t\)

In a chemical reaction, the rate of reaction is the derivative of the concentration \([\mathrm{C}](t)\) of the product of the reaction. What does $$\int_{t_{1}}^{t_{2}} \frac{d[\mathrm{C}]}{d t} d t$$ represent?

Evaluate the definite integral. \(\int_{0}^{1} \frac{d x}{(1+\sqrt{x})^{4}}\)

A honeybee population starts with 100 bees and increases at a rate of \(n^{\prime}(t)\) bees per week. What does the expression \(100+\int_{0}^{15} n^{\prime}(t) d t\) represent?

Verify that \(f(x)=\sin \sqrt[3]{x}\) is an odd function and use that fact to show that $$0 \leqslant \int_{-2}^{3} \sin \sqrt[3]{x} d x \leqslant 1$$

If oil leaks from a tank at a rate of \(r(t)\) gallons per minute at time \(t,\) what does \(\int_{0}^{120} r(t) d t\) represent?

Evaluate \(\int_{-2}^{2}(x+3) \sqrt{4-x^{2}} d x\) by writing it as a sum of two integrals and interpreting one of those integrals in terms of an area.

Suppose that a volcano is erupting and readings of the rate \(r(t)\) at which solid materials are spewed into the atmosphere are given in the table. The time \(t\) is measured in seconds and the units for \(r(t)\) are tonnes (metric tons) per second. $$\begin{array}{|c|c|c|c|c|c|c|}\hline t & {0} & {1} & {2} & {3} & {4} & {5} & {6} \\ \hline r(t) & {2} & {10} & {24} & {36} & {46} & {54} & {60} \\\ \hline\end{array}$$ (a) Give upper and lower estimates for the total quantity \(Q(6)\) of erupted materials after 6 seconds. (b) Use the Midpoint Rule to estimate \(Q(6)\)

Evaluate \(\int_{0}^{1} x \sqrt{1-x^{4}} d x\) by making a substitution and interpreting the resulting integral in terms of an area.

Water flows from the bottom of a storage tank at a rate of \(r(t)=200-4 t\) liters per minute, where 0\(\leqslant t \leqslant 50\) . Find the amount of water that flows from the tank during the first 10 minutes.

A bacteria population starts with 400 bacteria and grows at a rate of \(r(t)=(450.268) e^{1.12567 t}\) bacteria per hour. How many bacteria will there be after three hours?

Von Bertalanffy growth Many fish grow in a way that is described by the von Bertalanffy growth equation. For a fish that starts life with a length of 1 \(\mathrm{cm}\) and has a maximum length of \(30 \mathrm{cm},\) this equation predicts that the growth rate is 29\(e^{-a} \mathrm{cm} /\) year, where \(a\) is the age of the fish. How long will the fish be after 5 years?

An oil storage tank ruptures at time \(t=0\) and oil leaks from the tank at a rate of \(r(t)=100 e^{-0.01 t}\) liters per minute. How much oil leaks out during the first hour?