Chapter 11

Use a symbolic integration utility to find the indefinite integral. $$ \int\left(1+\frac{4}{t^{2}}\right)^{2}\left(\frac{1}{t^{3}}\right) d t $$

Find the indefinite integral and check your result by differentiation. $$ \int\left(\sqrt[3]{x}-\frac{1}{2 \sqrt[3]{x}}\right) d x $$

Use the Trapezoidal Rule with \(n=4\) to approximate the definite integral. $$ \int_{1}^{5} \frac{\sqrt{x-1}}{x} d x $$

Use a graphing utility to graph the region bounded by the graphs of the functions. Write the definite integrals that represent the area of the region. (Hint: Multiple integrals may be necessary.) $$ f(x)=x\left(x^{2}-3 x+3\right), g(x)=x^{2} $$

Evaluate the definite integral. $$ \int_{1}^{4} \sqrt{\frac{2}{x}} d x $$

Use a symbolic integration utility to find the indefinite integral. $$ \int\left(1+\frac{1}{t}\right)^{3}\left(\frac{1}{t^{2}}\right) d t $$

Find the indefinite integral and check your result by differentiation. $$ \int\left(\sqrt{x}+\frac{1}{2 \sqrt{x}}\right) d x $$

Use a computer or programmable calculator to approximate the definite integral using the Midpoint Rule and the Trapezoidal Rule for \(n=4\), \(8,12,16\), and 20. $$ \int_{0}^{4} \sqrt{2+3 x^{2}} d x $$

Use a graphing utility to graph the region bounded by the graphs of the functions. Write the definite integrals that represent the area of the region. (Hint: Multiple integrals may be necessary.) $$ y=\frac{4}{x}, y=x, x=1, x=4 $$

Evaluate the definite integral. $$ \int_{1}^{4} \frac{u-2}{\sqrt{u}} d u $$

Use a symbolic integration utility to find the indefinite integral. $$ \int\left(x^{3}+3 x+9\right)\left(x^{2}+1\right) d x $$

Find the indefinite integral and check your result by differentiation. $$ \int \sqrt[3]{x^{2}} d x $$

Use a computer or programmable calculator to approximate the definite integral using the Midpoint Rule and the Trapezoidal Rule for \(n=4\), \(8,12,16\), and 20. $$ \int_{0}^{2} \frac{5}{x^{3}+1} d x $$

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

Use a symbolic integration utility to find the indefinite integral. $$ \int\left(7-3 x-3 x^{2}\right)(2 x+1) d x $$

Find the indefinite integral and check your result by differentiation. $$ \int\left(\sqrt[4]{x^{3}}+1\right) d x $$

Use a graphing utility to graph the region bounded by the graphs of the functions, and find the area of the region. $$ f(x)=x^{2}-4 x, g(x)=0 $$

Evaluate the definite integral. $$ \int_{-1}^{0}\left(t^{1 / 3}-t^{2 / 3}\right) d t $$

Use a symbolic integration utility to find the indefinite integral. $$ \int \frac{e^{-x}}{1+e^{-x}} d x $$

Use formal substitution (as illustrated in Examples 5 and 6 ) to find the indefinite integral. $$ \int 12 x\left(6 x^{2}-1\right)^{3} d x $$

Find the indefinite integral and check your result by differentiation. $$ \int \frac{1}{x^{4}} d x $$

Use the Trapezoidal Rule with \(n=10\) to approximate the area of the region bounded by the graphs of the equations. $$ y=x \sqrt{\frac{4-x}{4+x}}, \quad y=0, \quad x=4 $$

Use a graphing utility to graph the region bounded by the graphs of the functions, and find the area of the region. $$ f(x)=3-2 x-x^{2}, g(x)=0 $$

Evaluate the definite integral. $$ \int_{0}^{4}\left(x^{1 / 2}+x^{1 / 4}\right) d x $$

Use a symbolic integration utility to find the indefinite integral. $$ \int \frac{3 e^{x}}{2+e^{x}} d x $$

Use formal substitution (as illustrated in Examples 5 and 6 ) to find the indefinite integral. $$ \int 3 x^{2}\left(1-x^{3}\right)^{2} d x $$

Find the indefinite integral and check your result by differentiation. $$ \int \frac{1}{4 x^{2}} d x $$

Evaluate the definite integral. $$ \int_{0}^{4} \frac{1}{\sqrt{2 x+1}} d x $$

Use formal substitution (as illustrated in Examples 5 and 6 ) to find the indefinite integral. $$ \int x^{2}\left(2-3 x^{3}\right)^{3 / 2} d x $$

Find the indefinite integral and check your result by differentiation. $$ \int \frac{2 x^{3}+1}{x^{3}} d x $$

Use a graphing utility to graph the region bounded by the graphs of the functions, and find the area of the region. $$ f(x)=-x^{2}+4 x+2, g(x)=x+2 $$

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

Use a symbolic integration utility to find the indefinite integral. $$ \int \frac{-e^{3 x}}{2-e^{3 x}} d x $$

Use formal substitution (as illustrated in Examples 5 and 6 ) to find the indefinite integral. $$ t \sqrt{t^{2}+1} d t $$

Find the indefinite integral and check your result by differentiation. $$ \int \frac{t^{2}+2}{t^{2}} d t $$

Numerical Approximation Use the Midpoint Rule and the Trapezoidal Rule with \(n=4\) to approximate \(\pi\) where \(\pi=\int_{0}^{1} \frac{4}{1+x^{2}} d x\) Then use a graphing utility to evaluate the definite integral. Compare all of your results.

Use integration to find the area of the triangular region having the given vertices. $$ \begin{aligned} &(0,0),(4,0),(4,4) \\ &(0,0),(4,0),(6,4) \end{aligned} $$

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

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int \frac{e^{2 x}+2 e^{x}+1}{e^{x}} d x $$

Use formal substitution (as illustrated in Examples 5 and 6 ) to find the indefinite integral. $$ \int \frac{x}{\sqrt{x^{2}+25}} d x $$

Use a symbolic integration utility to find the indefinite integral. $$ \int u\left(3 u^{2}+1\right) d u $$

Use integration to find the area of the triangular region having the given vertices. $$ (0,0),(4,0),(6,4) $$

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

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int\left(6 x+e^{x}\right) \sqrt{3 x^{2}+e^{x}} d x $$

Use formal substitution (as illustrated in Examples 5 and 6 ) to find the indefinite integral. $$ \int \frac{3}{\sqrt{2 x+1}} d x $$

Use a symbolic integration utility to find the indefinite integral. $$ \int \sqrt{x}(x+1) d x $$

Find the consumer and producer surpluses. $$ p_{1}(x)=50-0.5 x \quad p_{2}(x)=0.125 x $$

Evaluate the definite integral. $$ \int_{1}^{3} \frac{e^{3 / x}}{x^{2}} d x $$

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int e^{x} \sqrt{1-e^{x}} d x $$

Use formal substitution (as illustrated in Examples 5 and 6 ) to find the indefinite integral. $$ \int \frac{x^{2}+1}{\sqrt{x^{3}+3 x+4}} d x $$