Chapter 4

Evaluate the integral. $$\int \frac{e^{x}}{1+e^{2 x}} d x$$

Use the Fundamental Theorem if possible or estimate the integral using Riemann sums. $$\int_{0}^{2}(\sqrt{x}+1)^{2} d x$$

Use Theorem 4.2 to write the expression as a single integral. $$\int_{0}^{3} f(x) d x-\int_{2}^{3} f(x) d x$$

Use the given function values to estimate the area under the curve using left- endpoint and right-endpoint evaluation. $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline x & 1.0 & 1.2 & 1.4 & 1.6 & 1.8 & 2.0 & 2.2 & 2.4 & 2.6 \\ \hline f(x) & 0.0 & 0.4 & 0.6 & 0.8 & 1.2 & 1.4 & 1.2 & 1.4 & 1.0 \\ \hline \end{array}$$

Find the general antiderivative. $$\int \frac{3}{4 x^{2}+4} d x$$

Compute the sum and the limit of the sum as \(n \rightarrow \infty.\) $$\sum_{i=1}^{n} \frac{1}{n}\left[\left(\frac{2 i}{n}\right)^{2}+4\left(\frac{i}{n}\right)\right]$$

Evaluate the integral. $$\int \frac{e^{2 / x}}{x^{2}} d x$$

Use the Fundamental Theorem if possible or estimate the integral using Riemann sums. $$\int_{1}^{4} \frac{x^{2}}{x^{2}+4} d x$$

Graphically determine whether a Riemann sum with (a) left-endpoint, (b) midpoint and (c) right-endpoint evaluation points will be greater than or less than the area under the curve \(y=f(x)\) on \([a, b]\) \(f(x)\) is increasing and concave up on \([a, b]\)

Find the general antiderivative. $$\int\left(5 x-\frac{3}{e^{x}}\right) d x$$

Compute sums of the form \(\sum_{i=1}^{11} f\left(x_{i}\right) \Delta x\) for the given values. $$f(x)=x^{2}+4 x ; x=0.2,0.4,0.6,0.8,1.0 ; \Delta x=0.2 ; n=5$$

Evaluate the indicated integral. $$\int \frac{x^{3}}{\sqrt{1-x^{4}}} d x$$

Evaluate the integral. $$\int \frac{\sin \left(\ln x^{3}\right)}{x} d x$$

Use the Fundamental Theorem if possible or estimate the integral using Riemann sums. $$\int_{1}^{4} \frac{x^{2}+4}{x^{2}} d x$$

Graphically determine whether a Riemann sum with (a) left-endpoint, (b) midpoint and (c) right-endpoint evaluation points will be greater than or less than the area under the curve \(y=f(x)\) on \([a, b]\) \(f(x)\) is increasing and concave down on \([a, b]\)

Use Theorem 4.2 to write the expression as a single integral. $$\int_{-1}^{2} f(x) d x+\int_{2}^{3} f(x) d x$$

Find the general antiderivative. $$\int(2 \cos x-\sqrt{e^{2 x}}) d x$$

Compute sums of the form \(\sum_{i=1}^{11} f\left(x_{i}\right) \Delta x\) for the given values. $$f(x)=3 x+5 ; x=0.4,0.8,1.2,1.6,2.0 ; \Delta x=0.4 ; n=5$$

Evaluate the indicated integral. $$\int \frac{x^{2}}{1+x^{6}} d x$$

Evaluate the integral. $$\int_{0}^{1} \frac{x^{2}}{x^{3}-4} d x$$

Use the Fundamental Theorem if possible or estimate the integral using Riemann sums. $$\int_{0}^{\pi / 4} \frac{\sin x}{\cos ^{2} x} d x$$

Graphically determine whether a Riemann sum with (a) left-endpoint, (b) midpoint and (c) right-endpoint evaluation points will be greater than or less than the area under the curve \(y=f(x)\) on \([a, b]\) \(f(x)\) is decreasing and concave up on \([a, b]\)

Sketch the area corresponding to the integral. $$\int_{1}^{2}\left(x^{2}-x\right) d x$$

Find the general antiderivative. $$\int \frac{e^{x}}{e^{x}+3} d x$$

Compute sums of the form \(\sum_{i=1}^{11} f\left(x_{i}\right) \Delta x\) for the given values. $$\begin{array}{l} f(x)=4 x^{2}-2 ; x=2.1,2.2,2.3,2.4, \ldots, 3.0 \\ \Delta x=0.1 ; n=10 \end{array}$$

Evaluate the indicated integral. $$\int \frac{x^{5}}{1+x^{6}} d x$$

Evaluate the integral. $$\int_{0}^{1} \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} d x$$

Use the Fundamental Theorem if possible or estimate the integral using Riemann sums. $$\int_{0}^{\pi / 4} \frac{\tan x}{\sec ^{2} x} d x$$

Graphically determine whether a Riemann sum with (a) left-endpoint, (b) midpoint and (c) right-endpoint evaluation points will be greater than or less than the area under the curve \(y=f(x)\) on \([a, b]\) \(f(x)\) is decreasing and concave down on \([a, b]\)

Find the general antiderivative. $$\int \frac{\cos x}{\sin x} d x$$

Compute sums of the form \(\sum_{i=1}^{11} f\left(x_{i}\right) \Delta x\) for the given values. $$\begin{array}{l} f(x)=x^{3}+4 ; x=2.05,2.15,2.25,2.35, \ldots, 2.95 \\\ \Delta x=0.1 ; n=10 \end{array}$$

Evaluate the indicated integral. $$\int \frac{2 x+3}{x+7} d x$$

Evaluate the integral. $$\int_{0}^{1} \tan x d x$$

find the derivative \(f^{\prime}(x)\) \(f(x)=\int_{0}^{x}\left(t^{2}-3 t+2\right) d t\)

For the function \(f(x)=x^{2}\) on the interval [0,1] , by trial and error find evaluation points for \(n=2\) such that the Riemann sum equals the exact area of \(1 / 3\)

Find the general antiderivative. $$\int \frac{e^{x}+3}{e^{x}} d x$$

Suppose that a car has velocity 50 mph for 2 hours, velocity 60 mph for 1 hour, velocity 70 mph for 30 minutes and velocity 60 mph for 3 hours. Find the distance traveled.

Evaluate the indicated integral. $$\int \frac{x^{2}}{\sqrt[3]{x+3}} d x$$

Evaluate the integral. $$\int_{1}^{2} \frac{\ln x}{x} d x$$

find the derivative \(f^{\prime}(x)\) \(f(x)=\int_{2}^{x}\left(t^{2}-3 t-4\right) d t\)

For the function \(f(x)=\sqrt{x}\) on the interval \([0,1],\) by trial and error find evaluation points for \(n=2\) such that the Riemann sum equals the exact area of \(2 / 3\)

Find the general antiderivative. $$\int \frac{\left(e^{x}\right)^{2}-2}{e^{x}} d x$$

Suppose that a car has velocity 50 mph for 1 hour, velocity 40 mph for 1 hour, velocity 60 mph for 30 minutes and velocity 55 mph for 3 hours. Find the distance traveled.

Evaluate the indicated integral. $$\int \frac{1}{\sqrt{1+\sqrt{x}}} d x$$

Evaluate the integral. $$\int_{0}^{1} \frac{e^{x}-1}{e^{2 x}} d x$$

Use (a) Trapezoidal Rule and (b) Simpson's Rule to estimate \(\int_{0}^{2} f(x) d x\) from the given data. $$\begin{array}{|l|l|l|l|l|l|} \hline x & 0.0 & 0.25 & 0.5 & 0.75 & 1.0 \\ \hline f(x) & 4.0 & 4.6 & 5.2 & 4.8 & 5.0 \\ \hline \end{array}$$ $$\begin{array}{|l|l|l|l|l|} \hline x & 1.25 & 1.5 & 1.75 & 2.0 \\ \hline f(x) & 4.6 & 4.4 & 3.8 & 4.0 \\ \hline \end{array}$$

find the derivative \(f^{\prime}(x)\) \(f(x)=\int_{0}^{x^{2}}\left(e^{-t^{2}}+1\right) d t\)

Show that for right-endpoint evaluation on the interval \([a, b]\) with each sub interval of length \(\Delta x=(b-a) / n,\) the evaluation points are \(c_{i}=a+i \Delta x,\) for \(i=1,2, \ldots, n\)

Use the Integral Mean Value Theorem to estimate the value of the integral. $$\int_{\pi / 3}^{\pi / 2} 3 \cos x^{2} d x$$

Find the general antiderivative. $$\int x^{1 / 4}\left(x^{5 / 4}-4\right) d x$$