Chapter 11

Find the indefinite integral and check your result by differentiation. $$ \int v^{-1 / 2} d v $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ \begin{aligned} &y=x e^{-x^{2}}, y=0, x=0, x=1\\\ &\begin{gathered} 51 / 3 \end{gathered} \end{aligned} $$

Use the Log Rule to find the indefinite integral. $$ \int \frac{x^{2}}{x^{3}+1} d x $$

Find the indefinite integral and check the result by differentiation. $$ \int \frac{x-2}{\sqrt{x^{2}-4 x+3}} d x $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ y=\frac{e^{1 / x}}{x^{2}}, y=0, x=1, x=3 $$

Use the Log Rule to find the indefinite integral. $$ \frac{x}{x^{2}+4} d x $$

Find the indefinite integral and check the result by differentiation. $$ \int \frac{4 x+6}{\left(x^{2}+3 x+7\right)^{3}} d x $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(y)=\frac{1}{4} y, \quad[2,4] $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ y=\frac{8}{x}, y=x^{2}, y=0, x=1, x=4 $$

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

Use the Log Rule to find the indefinite integral. $$ \int \frac{x+3}{x^{2}+6 x+7} d x $$

Find the indefinite integral and check the result by differentiation. $$ \int 5 u \sqrt[3]{1-u^{2}} d u $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(y)=2 y, \quad[0,2] $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ y=\frac{1}{x}, y=x^{3}, x=\frac{1}{2}, x=1 $$

Evaluate the definite integral. $$ \int_{1}^{7} 3 v d v $$

Use the Log Rule to find the indefinite integral. $$ \int \frac{x^{2}+2 x+3}{x^{3}+3 x^{2}+9 x+1} d x $$

Find the indefinite integral and check the result by differentiation. $$ \int u^{3} \sqrt{u^{4}+2} d u $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(y)=y^{2}+1, \quad[0,4] $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ f(x)=e^{0.5 x}, g(x)=-\frac{1}{x}, x=1, x=2 $$

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

Use the Log Rule to find the indefinite integral. $$ \int \frac{1}{x \ln x} d x $$

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

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(y)=4 y-y^{2}, \quad[0,4] $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ f(x)=\frac{1}{x}, g(x)=-e^{x}, x=\frac{1}{2}, x=1 $$

Evaluate the definite integral. $$ \int_{2}^{5}(-3 x+4) d x $$

Use the Log Rule to find the indefinite integral. $$ \int \frac{1}{x(\ln x)^{2}} d x $$

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

Use the Trapezoidal Rule with \(n=8\) to approximate the definite integral. Compare the result with the exact value and the approximation obtained with \(n=8\) and the Midpoint Rule. Which approximation technique appears to be better? Let \(f\) be continuous on \([a, b]\) and let \(n\) be the number of equal subintervals (see figure). Then the Trapezoidal Rule for approximating \(\int_{a}^{b} f(x) d x\) is \(\frac{b-a}{2 n}\left[f\left(x_{0}\right)+2 f\left(x_{1}\right)+\cdots+2 f\left(x_{n-1}\right)+f\left(x_{n}\right)\right]\). $$ \int_{0}^{2} x^{3} d x $$

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

Use the Log Rule to find the indefinite integral. $$ \int \frac{e^{-x}}{1-e^{-x}} d x $$

Find the indefinite integral and check the result by differentiation. $$ \int \frac{-3}{\sqrt{2 t+3}} d t $$

Find the indefinite integral and check your result by differentiation. $$ \int(x+3) d x $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ f(y)=y(2-y), g(y)=-y $$

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

Use the Log Rule to find the indefinite integral. $$ \int \frac{e^{x}}{1+e^{x}} d x $$

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

Find the indefinite integral and check your result by differentiation. $$ \int(5-x) d x $$

Use the Trapezoidal Rule with \(n=4\) to approximate the definite integral. $$ \int_{0}^{2} \frac{1}{x+1} d x $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ f(y)=\sqrt{y}, y=9, x=0 $$

Evaluate the definite integral. $$ \int_{0}^{3}(x-2)^{3} d x $$

Use a symbolic integration utility to find the indefinite integral. $$ \int \frac{x^{3}}{\sqrt{1-x^{4}}} d x $$

Find the indefinite integral and check your result by differentiation. $$ \int\left(x^{3}+2\right) d x $$

Use the Trapezoidal Rule with \(n=4\) to approximate the definite integral. $$ \int_{0}^{4} \sqrt{1+x^{2}} d x $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ f(y)=y^{2}+1, g(y)=4-2 y $$

Evaluate the definite integral. $$ \int_{2}^{2}(x-3)^{4} d x $$

Use a symbolic integration utility to find the indefinite integral. $$ \int \frac{3 x}{\sqrt{1-4 x^{2}}} d x $$

Find the indefinite integral and check your result by differentiation. $$ \int\left(x^{3}-4 x+2\right) d x $$

Use the Trapezoidal Rule with \(n=4\) to approximate the definite integral. $$ \int_{-1}^{1} \frac{1}{x^{2}+1} d x $$

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.) $$ f(x)=2 x, g(x)=4-2 x, h(x)=0 $$

Evaluate the definite integral. $$ \int_{-1}^{1}(\sqrt[3]{t}-2) d t $$