Chapter 7

Use the Fundamental Theorem to find the area under \(f(x)=x^{2}\) between \(x=1\) and \(x=4\).

Find an antiderivative. $$ g(\theta)=\sin \theta-2 \cos \theta $$

Find the integrals in problems. Check your answers by differentiation. $$ \int \sin ^{3} \alpha \cos \alpha d \alpha $$

The concentration, \(C\), in \(\mathrm{ng} / \mathrm{ml}\), of a drug in the blood as a function of the time, \(t\), in hours since the drug was administered is given by \(C=15 t e^{-0.2 t}\). The area under the concentration curve is a measure of the overall effect of the drug on the body, called the bioavailability. Find the bioavailability of the drug between \(t=0\) and \(t=3\).

Write the definite integral for the area under the graph of \(f(x)=6 x^{2}+1\) between \(x=0\) and \(x=2\). Use the Fundamental Theorem of Calculus to evaluate it.

Find an antiderivative \(F(x)\) with \(F^{\prime}(x)=\) \(f(x)\) and \(F(0)=0\). Is there only one possible solution? $$ f(x)=3 $$

Find the integrals in problems. Check your answers by differentiation. $$ \int x \sin \left(4 x^{2}\right) d x $$

During a surge in the demand for electricity, the rate, \(r\), at which energy is used can be approximated by $$ r=t e^{-a t} $$ where \(t\) is the time in hours and \(a\) is a positive constant. (a) Find the total energy, \(E\), used in the first \(T\) hours. Give your answer as a function of \(a .\) (b) What happens to \(E\) as \(T \rightarrow \infty\) ?

Use the Fundamental Theorem to find the average value of \(f(x)=x^{2}+1\) on the interval \(x=0\) to \(x=10\). Illustrate your answer on a graph of \(f(x)\).

Find an antiderivative \(F(x)\) with \(F^{\prime}(x)=\) \(f(x)\) and \(F(0)=0\). Is there only one possible solution? $$ f(x)=-7 x $$

Find the integrals in problems. Check your answers by differentiation. $$ \int x e^{3 x^{2}} d x $$

Derive the formula (called a reduction formula): $$ \int x^{n} e^{x} d x=x^{n} e^{x}-n \int x^{n-1} e^{x} d x $$

Use the Fundamental Theorem of Calculus to find the average value of \(f(x)=e^{0.5 x}\) between \(x=0\) and \(x=3\). Show the average value on a graph of \(f(x)\).

Find an antiderivative \(F(x)\) with \(F^{\prime}(x)=\) \(f(x)\) and \(F(0)=0\). Is there only one possible solution? $$ f(x)=2+4 x+5 x^{2} $$

Find the integrals in problems. Check your answers by differentiation. $$ \int x e^{3 x^{2}} d x $$

Find the exact area of the region bounded by the \(x\) -axis and the graph of \(y=x^{3}-x\).

Find an antiderivative \(F(x)\) with \(F^{\prime}(x)=\) \(f(x)\) and \(F(0)=0\). Is there only one possible solution? $$ f(x)=x^{2} $$

Find the integrals in problems. Check your answers by differentiation. $$ \int x \sqrt{3 x^{2}+4} d x $$

Use the Fundamental Theorem to determine the value of \(b\) if the area under the graph of \(f(x)=4 x\) between \(x=1\) and \(x=b\) is equal to \(240 .\) Assume \(b>1\).

Find the integrals in problems. Check your answers by differentiation. $$ \int \frac{q}{5 q^{2}+8} d q $$

Use the Fundamental Theorem to determine the value of \(b\) if the area under the graph of \(f(x)=8 x\) between \(x=1\) and \(x=b\) is equal to \(192 .\) Assume \(b>1\).

Find an antiderivative \(F(x)\) with \(F^{\prime}(x)=\) \(f(x)\) and \(F(0)=0\). Is there only one possible solution? $$ f(x)=e^{x} $$

Find the integrals in problems. Check your answers by differentiation. $$ \int \frac{(\ln z)^{2}}{z} d z $$

Find the indefinite integrals. $$ \int(5 x+7) d x $$

Use the Fundamental Theorem to determine the value of \(b\) if the area under the graph of \(f(x)=x^{2}\) between \(x=0\) and \(x=b\) is equal to \(100 .\) Assume \(b>0\).

Find the integrals in problems. Check your answers by differentiation. $$ \int \frac{y}{y^{2}+4} d y $$

Find the indefinite integrals. $$ \int 9 x^{2} d x $$

Oil is leaking out of a ruptured tanker at the rate of \(r(t)=50 e^{-0.02 t}\) thousand liters per minute. (a) At what rate, in liters per minute, is oil leaking out at \(t=0 ?\) At \(t=60\) ? (b) How many liters leak out during the first hour?

Find the integrals in problems. Check your answers by differentiation. $$ \int \frac{e^{t}+1}{e^{t}+t} d t $$

Find the indefinite integrals. $$ \int 6 x^{2} d x $$

Find the integrals in problems. Check your answers by differentiation. $$ \int \frac{e^{\sqrt{y}}}{\sqrt{y}} d y $$

Find the indefinite integrals. $$ \int t^{12} d t $$

(a) Graph \(f(x)=e^{-x^{2}}\) and shade the area represented by the improper integral \(\int_{-\infty}^{\infty} e^{-x^{2}} d x\) (b) Find \(\int_{-a}^{a} e^{-x^{2}} d x\) for \(a=1, a=2, a=3, a=5\). (c) The improper integral \(\int_{-\infty}^{\infty} e^{-x^{2}} d x\) converges to a finite value. Use your answers from part (b) to estimate that value.

Find the integrals in problems. Check your answers by differentiation. $$ \int \frac{\cos \sqrt{x}}{\sqrt{x}} d x $$

Find the indefinite integrals. $$ \int(x+1)^{2} d x $$

Graph \(y=1 / x^{2}\) and \(y=1 / x^{3}\) on the same axes. Which do you think is larger: \(\int_{1}^{\infty} 1 / x^{2} d x\) or \(\int_{1}^{\infty} 1 / x^{3} d x\) ? Why?

Find the integrals in problems. Check your answers by differentiation. $$ \int \frac{1+e^{x}}{\sqrt{x+e^{x}}} d x $$

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

In this problem, you will show that the following improper integral converges to 1 . $$\int_{1}^{\infty} \frac{1}{x^{2}} d x$$ (a) Use the Fundamental Theorem to find \(\int_{1}^{b} 1 / x^{2} d x\). Your answer will contain \(b\). (b) Now take the limit as \(b \rightarrow \infty\). What does this tell you about the improper integral?

Find the integrals in problems. Check your answers by differentiation. $$ \int \frac{e^{t}}{e^{t}+1} d t $$

Find the indefinite integrals. $$ \int\left(t^{2}+5 t+1\right) d t $$

Decide if the improper integral \(\int_{0}^{\infty} e^{-2 t} d t\) converges, and if so, to what value, by the following method. (a) Evaluate \(\int_{0}^{b} e^{-2 t} d t\) for \(b=3,5,7,10\). What do you observe? Make a guess about the convergence of the improper integral. (b) Find \(\int_{0}^{b} e^{-2 t} d t\) using the Fundamental Theorem. Your answer will contain \(b\). (c) Take a limit as \(b \rightarrow \infty\). Does your answer confirm your guess?

Find the integrals in problems. Check your answers by differentiation. $$ \int \frac{x+1}{x^{2}+2 x+19} d x $$

Find the indefinite integrals. $$ \int 5 e^{z} d z $$

(a) Evaluate \(\int_{0}^{b} x e^{-x / 10} d x\) for \(b=10,50,100,200\). (b) Assuming that it converges, estimate the value of \(\int_{0}^{\infty} x e^{-x / 10} d x\).

Find the integrals in problems. Check your answers by differentiation. $$ \int \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} d x $$

Find the indefinite integrals. $$ \int\left(t^{3}+6 t^{2}\right) d t $$

If appropriate, evaluate the following integrals by substitution. If substitution is not appropriate, say so, and do not evaluate. (a) \(\int x \sin \left(x^{2}\right) d x\) (b) \(\int x^{2} \sin x d x\) (c) \(\int \frac{x^{2}}{1+x^{2}} d x\) (d) \(\int \frac{x}{\left(1+x^{2}\right)^{2}} d x\) (e) \(\int x^{3} e^{x^{2}} d x\) (f) \(\int \frac{\sin x}{2+\cos x} d x\)

Find the indefinite integrals. $$ \int\left(x^{5}-12 x^{3}\right) d x $$

Find the exact area below the curve \(y=x^{3}(1-x)\) and above the \(x\) -axis.