Chapter 5

In the following exercises, \(f(x) \geq 0\) for \(a \leq x \leq b\) . Find the area under the graph of \(f(x)\) between the given values \(a\) and \(b\) by integrating. $$f(x)=2^{-x} ; a=3, b=4$$

Find the area under the graph of the function \(f(x)=x e^{-x^{2}}\) between \(x=0\) and \(x=5.\)

Compute the integral of \(f(x)=x e^{-x^{2}}\) and find the smallest value of \(N\) such that the area under the graph \(f(x)=x e^{-x^{2}}\) between \(x=N\) and \(x=N+10\) is, at most, 0.01.

Find the limit, as \(N\) tends to infinity, of the area under the graph of \(f(x)=x e^{-x^{2}}\) between \(x=0\) and \(x=5\) .

Show that \(\int_{a}^{b} \frac{d t}{t}=\int_{1 / b}^{1 / a} \frac{d t}{t}\) when \(0 < a \leq b.\)

Suppose that \(f(x)>0\) for all \(x\) and that \(f\) and \(g\) are differentiable. Use the identity \(f^{g}=e^{g \ln f}\) and the chain rule to find the derivative of \(f^{g} .\)

Show that if \(c>0,\) then the integral of 1\(/ x\) from ac to \(b c(0 < a < b)\) is the same as the integral of 1\(/x\) from \(a\) to \(b\) .

Show that if \(c>0\), then the integral of \(1 / x\) from \(a c\) to \(b c(0

The following exercises are intended to derive the fundamental properties of the natural log starting from the definition \(\ln (x)=\int_{1}^{x} \frac{d t}{t}, \quad\) using properties of the definite integral and making no further assumptions. Use the identity \(\ln (x)=\int_{1}^{x} \frac{d t}{t}\) to derive the identity \(\ln \left(\frac{1}{x}\right)=-\ln x.\)

The following exercises are intended to derive the fundamental properties of the natural log starting from the definition \(\ln (x)=\int_{1}^{x} \frac{d t}{t}, \quad\) using properties of the definite integral and making no further assumptions. Use a change of variable in the integral \(\int_{1}^{x y} \frac{1}{t} d t\) to show that \(\ln x y=\ln x+\ln y\) for \(x, y>0.\)

The following exercises are intended to derive the fundamental properties of the natural log starting from the definition \(\ln (x)=\int_{1}^{x} \frac{d t}{t},\) using properties of the definite integral and making no further assumptions. Use the identity \(\ln x=\int_{1}^{x} \frac{d t}{x}\) to show that \(\ln (x)\) is an increasing function of \(x\) on \([0, \infty),\) and use the previous exercises to show that the range of \(\ln (x)\) is \((-\infty, \infty)\). Without any further assumptions, conclude that \(\ln (x)\) has an inverse function defined on \((-\infty, \infty)\).

The following exercises are intended to derive the fundamental properties of the natural log starting from the definition \(\ln (x)=\int_{1}^{x} \frac{d t}{t}, \quad\) using properties of the definite integral and making no further assumptions. Pretend, for the moment, that we do not know that \(e^{x}\) is the inverse function of \(\ln (x),\) but keep in mind that \(\ln (x)\) has an inverse function defined on \((-\infty, \infty) .\) Call it \(E\) . Use the identity \(\ln x y=\ln x+\ln y\) to deduce that \(E(a+b)=E(a) E(b)\) for any real numbers \(a, b.\)

Pretend, for the moment, that we do not know that \(e^{x}\) is the inverse function of \(\ln (x),\) but keep in mind that \(\ln (x)\) has an inverse function defined on \((-\infty, \infty)\). Call it \(E\). Use the identity \(\ln x y=\ln x+\ln y\) to deduce that \(E(a+b)=E(a) E(b)\) for any real numbers \(a, b\).

The following exercises are intended to derive the fundamental properties of the natural log starting from the definition \(\ln (x)=\int_{1}^{x} \frac{d t}{t}, \quad\) using properties of the definite integral and making no further assumptions. Pretend, for the moment, that we do not know that \(e^{x}\) is the inverse function of \(\ln x,\) but keep in mind that \(\ln x\) has an inverse function defined on \((-\infty, \infty) .\) Call it \(E\) Show that \(E^{\prime}(t)=E(t).\)

Pretend, for the moment, that we do not know that \(e^{x}\) is the inverse function of \(\ln x,\) but keep in mind that \(\ln x\) has an inverse function defined on \((-\infty, \infty) .\) Call it \(E\). Show that \(E^{\prime}(t)=E(t)\).

The following exercises are intended to derive the fundamental properties of the natural log starting from the definition \(\ln (x)=\int_{1}^{x} \frac{d t}{t}, \quad\) using properties of the definite integral and making no further assumptions. The sine integral, defined as \(S(x)=\int_{0}^{x} \frac{\sin t}{t} d t\) is an important quantity in engineering. Although it does not have a simple closed formula, it is possible to estimate its behavior for large \(x\) . Show that for \(k \geq 1,|S(2 \pi k)-S(2 \pi(k+1))| \leq \frac{1}{k(2 k+1) \pi}\) \((\text {Hint} : \sin (t+\pi)=-\sin t)\)

The sine integral, defined as \(S(x)=\int_{0}^{x} \frac{\sin t}{t} d t\) is an important quantity in engineering. Although it does not have a simple closed formula, it is possible to estimate its behavior for large \(x\). Show that for \(k \geq 1,|S(2 \pi k)-S(2 \pi(k+1))| \leq \frac{1}{k(2 k+1) \pi}\) \((\) Hint \(: \sin (t+\pi)=-\sin t)\)

[T] Compute the right endpoint estimates \(R_{50}\) and \(R_{100}\) of \(\int_{-3}^{5} \frac{1}{2 \sqrt{2 \pi}} e^{-(x-1)^{2} / 8}\)

In the following exercises, evaluate each integral in terms of an inverse trigonometric function. $$\int_{0}^{\sqrt{3} / 2} \frac{d x}{\sqrt{1-x^{2}}}$$

In the following exercises, evaluate each integral in terms of an inverse trigonometric function. $$\int_{-1 / 2}^{1 / 2} \frac{d x}{\sqrt{1-x^{2}}}$$

In the following exercises, evaluate each integral in terms of an inverse trigonometric function. $$\int_{\sqrt{3}}^{1} \frac{d x}{\sqrt{1+x^{2}}}$$

In the following exercises, evaluate each integral in terms of an inverse trigonometric function. $$\int_{1 / \sqrt{3}}^{\sqrt{3}} \frac{d x}{1+x^{2}}$$

In the following exercises, evaluate each integral in terms of an inverse trigonometric function. $$\int_{1}^{\sqrt{2}} \frac{d x}{|x| \sqrt{x^{2}-1}}$$

In the following exercises, evaluate each integral in terms of an inverse trigonometric function. $$\int_{1}^{2 \sqrt{3}} \frac{d x}{|x| \sqrt{x^{2}-1}}$$

In the following exercises, find each indefinite integral, using appropriate substitutions. $$ \int \frac{d x}{\sqrt{9-x^{2}}}$$

In the following exercises, find each indefinite integral, using appropriate substitutions. $$\int \frac{d x}{\sqrt{1-16 x^{2}}}$$

In the following exercises, find each indefinite integral, using appropriate substitutions. $$\int \frac{d x}{9+x^{2}}$$

In the following exercises, find each indefinite integral, using appropriate substitutions. $$\int \frac{d x}{25+16 x^{2}}$$

In the following exercises, find each indefinite integral, using appropriate substitutions. $$\int \frac{d x}{|x| \sqrt{x^{2}-9}}$$

In the following exercises, find each indefinite integral, using appropriate substitutions. $$\int \frac{d x}{|x| \sqrt{4 x^{2}-16}}$$

Explain the relationship \(-\cos ^{-1} t+C=\int \frac{d t}{\sqrt{1-t^{2}}}=\sin ^{-1} t+C .\) Is it true, in general, that \(\cos ^{-1} t=-\sin ^{-1} t ?\)

Explain the relationship \(\sec ^{-1} t+C=\int \frac{d t}{|t| \sqrt{t^{2}-1}}=-\csc ^{-1} t+C .\) Is it true, in general, that \(\sec ^{-1} t=-\csc ^{-1} t ?\)

Explain what is wrong with the following integral: $$\int_{1}^{2} \frac{d t}{\sqrt{1-t^{2}}}$$

Explain what is wrong with the following integral: $$\int_{-1}^{1} \frac{d t}{|t| \sqrt{t^{2}-1}}$$

In the following exercises, solve for the antiderivative \(\int f\) of \(f\) with \(C=0,\) then use a calculator to graph \(f\) and the antiderivative over the given interval \([a, b] .\) Identify a value of \(C\) such that adding \(C\) to the antiderivative recovers the definite integral \(F(x)=\int_{a}^{x} f(t) d t .\) $$[\mathbf{T}] \int \frac{1}{\sqrt{9-x^{2}}} d x \text { over }[-3,3]$$

In the following exercises, solve for the antiderivative \(\int f\) of \(f\) with \(C=0,\) then use a calculator to graph \(f\) and the antiderivative over the given interval \([a, b] .\) Identify a value of \(C\) such that adding \(C\) to the antiderivative recovers the definite integral \(F(x)=\int_{a}^{x} f(t) d t .\) $$[\mathbf{T}] \int \frac{9}{9+x^{2}} d x \text { over } [-6,6]$$

In the following exercises, solve for the antiderivative \(\int f\) of \(f\) with \(C=0,\) then use a calculator to graph \(f\) and the antiderivative over the given interval \([a, b]\). Identify a value of \(C\) such that adding \(C\) to the antiderivative recovers the definite integral \(F(x)=\int_{a}^{x} f(t) d t\). [T] \(\int \frac{\cos x}{4+\sin ^{2} x} d x\) over [-6,6]

In the following exercises, solve for the antiderivative \(\int f\) of \(f\) with \(C=0,\) then use a calculator to graph \(f\) and the antiderivative over the given interval \([a, b] .\) Identify a value of \(C\) such that adding \(C\) to the antiderivative recovers the definite integral \(F(x)=\int_{a}^{x} f(t) d t .\) $$[\mathbf{T}] \int \frac{e^{x}}{1+e^{2 x}} d x \text { over }[-6,6]$$

In the following exercises, compute the antiderivative using appropriate substitutions. $$\int \frac{\sin ^{-1} t d t}{\sqrt{1-t^{2}}}$$

In the following exercises, compute the antiderivative using appropriate substitutions. $$\int \frac{d t}{\sin ^{-1} t \sqrt{1-t^{2}}}$$

In the following exercises, compute the antiderivative using appropriate substitutions. $$\int \frac{\tan ^{-1}(2 t)}{1+4 t^{2}} d t$$

In the following exercises, compute the antiderivative using appropriate substitutions. $$\int \frac{t \tan ^{-1}\left(t^{2}\right)}{1+t^{4}} d t$$

In the following exercises, compute the antiderivative using appropriate substitutions. \(\int \frac{\sec ^{-1}\left(\frac{t}{2}\right)}{|t| \sqrt{t^{2}-4}} d t\)

In the following exercises, compute the antiderivative using appropriate substitutions. $$\int \frac{t \sec ^{-1}\left(t^{2}\right)}{t^{2} \sqrt{t^{4}-1}} d t$$

In the following exercises, use a calculator to graph the antiderivative \(\int f\) with \(C=0\) over the given interval \([a, b]\) . Approximate a value of \(C,\) if possible, such that adding \(C\) to the antiderivative gives the same value as the definite integral \(F(x)=\int_{a}^{x} f(t) d t\). $$[\mathbf{T}] \int \frac{1}{(2 x+2) \sqrt{x}} d x \text { over }[0,6]$$

In the following exercises, use a calculator to graph the antiderivative \(\int f\) with \(C=0\) over the given interval \([a, b]\) . Approximate a value of \(C,\) if possible, such that adding \(C\) to the antiderivative gives the same value as the definite integral \(F(x)=\int_{a}^{x} f(t) d t\). $$[\mathbf{T}] \int \frac{(\sin x+x \cos x)}{1+x^{2} \sin ^{2} x} d x \text { over }[-6,6]$$

In the following exercises, use a calculator to graph the antiderivative \(\int f\) with \(C=0\) over the given interval \([a, b]\) . Approximate a value of \(C,\) if possible, such that adding \(C\) to the antiderivative gives the same value as the definite integral \(F(x)=\int_{a}^{x} f(t) d t\). $$[\mathbf{T}] \int \frac{2 e^{-2 x}}{\sqrt{1-e^{-4 x}}} d x \text { over }[0,2]$$

In the following exercises, compute each integral using appropriate substitutions. $$\int \frac{e^{t}}{1+e^{2 t}} d t$$

In the following exercises, compute each integral using appropriate substitutions. $$\int \frac{d t}{t \sqrt{1-\ln ^{2} t}}$$

In the following exercises, compute each integral using appropriate substitutions. $$\int \frac{d t}{t\left(1+\ln ^{2} t\right)}$$