Chapter 4

In Exercises \(1-6,\) evaluate the function. If the value is not a rational number, round your answer to three decimal places. (a) \(\sinh 3\) (b) \(\tanh (-2)\)

Find the integral. $$ \int \frac{5}{\sqrt{9-x^{2}}} d x $$

In Exercises 1 and 2, use Example 1 as a model to evaluate the limit $$\lim _{n \rightarrow \infty} \sum_{i=1}^{n} f\left(c_{i}\right) \Delta x_{i}$$ over the region bounded by the graphs of the equations. $$ \begin{array}{l} f(x)=\sqrt{x}, \quad y=0, \quad x=0, \quad x=3 \\ \left(\text { Hint: } \text { Let } c_{i}=3 i^{2} / n^{2} .\right) \end{array} $$

Graphical Reasoning In Exercises \(1-4,\) use a graphing utility to graph the integrand. Use the graph to determine whether the definite integral is positive, negative, or zero. $$ \int_{0}^{\pi} \frac{4}{x^{2}+1} d x $$

Complete the table by identifying \(u\) and \(d u\) for the integral. $$ \int f(g(x)) g^{\prime}(x) d x \quad \underline{u=g(x)} \quad \underline{d u=g^{\prime}(x) d x} $$ $$ \int\left(5 x^{2}+1\right)^{2}(10 x) d x $$

Find the indefinite integral. $$ \int \frac{5}{x} d x $$

In Exercises \(1-6,\) use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of \(n .\) Round your answers to four decimal places and compare your results with the exact value of the definite integral. $$ \int_{0}^{2} x^{2} d x, \quad n=4 $$

In Exercises \(1-6,\) find the sum. Use the summation capabilities of a graphing utility to verify your result. $$ \sum_{i=1}^{5}(2 i+1) $$

Verify the statement by showing that the derivative of the right side equals the integrand of the left side. $$ \int(x-2)(x+2) d x=\frac{1}{3} x^{3}-4 x+C $$

Evaluate the function. If the value is not a rational number, round your answer to three decimal places. (a) \(\cosh 0\) (b) \(\operatorname{sech} 1\)

Find the integral. $$ \int \frac{4}{1+9 x^{2}} d x $$

In Exercises 1 and 2, use Example 1 as a model to evaluate the limit $$\lim _{n \rightarrow \infty} \sum_{i=1}^{n} f\left(c_{i}\right) \Delta x_{i}$$ over the region bounded by the graphs of the equations. $$ \begin{array}{l} f(x)=2 \sqrt[3]{x}, \quad y=0, \quad x=0, \quad x=1 \\ \text { (Hint: Let } \left.c_{i}=i^{3} / n^{3} .\right) \end{array} $$

Graphical Reasoning use a graphing utility to graph the integrand. Use the graph to determine whether the definite integral is positive, negative, or zero. $$ \int_{0}^{\pi} \cos x d x $$

Complete the table by identifying \(u\) and \(d u\) for the integral. $$ \int f(g(x)) g^{\prime}(x) d x \quad \underline{u=g(x)} \quad \underline{d u=g^{\prime}(x) d x} $$ $$ \int x^{2} \sqrt{x^{3}+1} d x $$

Find the indefinite integral. $$ \int \frac{1}{x-5} d x $$

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of \(n .\) Round your answers to four decimal places and compare your results with the exact value of the definite integral. $$ \int_{1}^{2} \frac{2}{x^{2}} d x, \quad n=4 $$

In Exercises \(1-6,\) find the sum. Use the summation capabilities of a graphing utility to verify your result. $$ \sum_{k=3}^{6} k(k-2) $$

Verify the statement by showing that the derivative of the right side equals the integrand of the left side. $$ \int \frac{x^{2}-1}{x^{3 / 2}} d x=\frac{2\left(x^{2}+3\right)}{3 \sqrt{x}}+C $$

Evaluate the function. If the value is not a rational number, round your answer to three decimal places. (a) \(\operatorname{csch}(\ln 2)\) (b) \(\operatorname{coth}(\ln 5)\)

Find the integral. $$ \int \frac{1}{x \sqrt{4 x^{2}-1}} d x $$

In Exercises 3-8, evaluate the definite integral by the limit definition. $$ \int_{4}^{10} 6 d x $$

Graphical Reasoning use a graphing utility to graph the integrand. Use the graph to determine whether the definite integral is positive, negative, or zero. $$ \int_{-2}^{2} x \sqrt{x^{2}+1} d x $$

Complete the table by identifying \(u\) and \(d u\) for the integral. $$ \int f(g(x)) g^{\prime}(x) d x \quad \underline{u=g(x)} \quad \underline{d u=g^{\prime}(x) d x} $$ $$ \int \frac{x}{\sqrt{x^{2}+1}} d x $$

Find the indefinite integral. $$ \int \frac{1}{3-2 x} d x $$

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of \(n .\) Round your answers to four decimal places and compare your results with the exact value of the definite integral. $$ \int_{0}^{2} x^{3} d x, \quad n=8 $$

In Exercises \(1-6,\) find the sum. Use the summation capabilities of a graphing utility to verify your result. $$ \sum_{k=0}^{4} \frac{1}{k^{2}+1} $$

Find the general solution of the differential equation and check the result by differentiation. $$ \frac{d y}{d t}=3 t^{2} $$

Evaluate the function. If the value is not a rational number, round your answer to three decimal places. (a) \(\sinh ^{-1} 0\) (b) \(\tanh ^{-1} 0\)

Find the integral. $$ \int \frac{1}{4+(x-1)^{2}} d x $$

In Exercises 3-8, evaluate the definite integral by the limit definition. $$ \int_{-2}^{3} x d x $$

Graphical Reasoning use a graphing utility to graph the integrand. Use the graph to determine whether the definite integral is positive, negative, or zero. $$ \int_{-2}^{2} x \sqrt{2-x} d x $$

Complete the table by identifying \(u\) and \(d u\) for the integral. $$ \int f(g(x)) g^{\prime}(x) d x \quad \underline{u=g(x)} \quad \underline{d u=g^{\prime}(x) d x} $$ $$ \int \sec 2 x \tan 2 x d x $$

Find the indefinite integral. $$ \int \frac{x^{2}}{3-x^{3}} d x $$

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of \(n .\) Round your answers to four decimal places and compare your results with the exact value of the definite integral. $$ \int_{0}^{8} \sqrt[3]{x} d x, \quad n=8 $$

In Exercises \(1-6,\) find the sum. Use the summation capabilities of a graphing utility to verify your result. $$ \sum_{j=3}^{5} \frac{1}{j} $$

Find the general solution of the differential equation and check the result by differentiation. $$ \frac{d r}{d \theta}=\pi $$

Evaluate the function. If the value is not a rational number, round your answer to three decimal places. (a) \(\cosh ^{-1} 2\) (b) \(\operatorname{sech}^{-1} \frac{2}{3}\)

Find the integral. $$ \int \frac{x^{3}}{x^{2}+1} d x $$

In Exercises 3-8, evaluate the definite integral by the limit definition. $$ \int_{-1}^{1} x^{3} d x $$

In Exercises \(5-18,\) evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. $$ \int_{0}^{1} 2 x d x $$

Complete the table by identifying \(u\) and \(d u\) for the integral. $$ \int f(g(x)) g^{\prime}(x) d x \quad \underline{u=g(x)} \quad \underline{d u=g^{\prime}(x) d x} $$ $$ \int \tan ^{2} x \sec ^{2} x d x $$

Find the indefinite integral. $$ \int \frac{x^{2}-4}{x} d x $$

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of \(n .\) Round your answers to four decimal places and compare your results with the exact value of the definite integral. $$ \int_{1}^{2} \frac{1}{(x+1)^{2}} d x, \quad n=4 $$

Find the general solution of the differential equation and check the result by differentiation. $$ \frac{d y}{d x}=x^{3 / 2} $$

Evaluate the function. If the value is not a rational number, round your answer to three decimal places. (a) \(\operatorname{csch}^{-1} 2\) (b) \(\operatorname{coth}^{-1} 3\)

Find the integral. $$ \int \frac{x^{4}-1}{x^{2}+1} d x $$

In Exercises 3-8, evaluate the definite integral by the limit definition. $$ \int_{1}^{3} 3 x^{2} d x $$

Evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. $$ \int_{2}^{7} 3 d v $$

Complete the table by identifying \(u\) and \(d u\) for the integral. $$ \int f(g(x)) g^{\prime}(x) d x \quad \underline{u=g(x)} \quad \underline{d u=g^{\prime}(x) d x} $$ $$ \int \frac{\cos x}{\sin ^{3} x} d x $$

Find the indefinite integral. $$ \int \frac{x}{\sqrt{9-x^{2}}} d x $$