Chapter 4

Find the general antiderivative. $$\int\left(2 x^{-2}+\frac{1}{\sqrt{x}}\right) d x$$

Evaluate the indicated integral. $$\int x^{2} \cos x^{3} d x$$

Use the properties of logarithms to rewrite the expression as a single term. $$\ln \sqrt{2}+3 \ln 2$$

Use a computer or calculator to compute the Midpoint, Trapezoidal and Simpson's Rule approximations with \(n=10, n=20\) and \(n=50 .\) Compare these values to the approximation given by your calculator or computer. $$\int_{0}^{2} e^{-x^{2}} d x$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{0}^{\pi / 4} \sec x \tan x d x$$

Evaluate the integral by computing the limit of Riemann sums. $$\int_{1}^{3}\left(x^{2}-3\right) d x$$

Use summation rules to compute the sum. $$\sum_{i=1}^{70}(3 i-1)$$

Find the general antiderivative. $$\int \frac{x^{1 / 3}-3}{x^{2 / 3}} d x$$

Evaluate the indicated integral. $$\int \sin x(\cos x+3)^{3 / 4} d x$$

Use the properties of logarithms to rewrite the expression as a single term. $$\ln 8-2 \ln 2$$

Use a computer or calculator to compute the Midpoint, Trapezoidal and Simpson's Rule approximations with \(n=10, n=20\) and \(n=50 .\) Compare these values to the approximation given by your calculator or computer. $$\int_{0}^{3} e^{-x^{2}} d x$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{0}^{\pi / 4} \sec ^{2} x d x$$

Approximate the area under the curve on the given interval using \(n\) rectangles and the evaluation rules (a) left endpoint (b) midpoint (e) right endpoint. $$y=x^{3}-1 \text { on }[-1,1], n=100$$

Evaluate the integral by computing the limit of Riemann sums. $$\int_{-2}^{2}\left(x^{2}-1\right) d x$$

Use summation rules to compute the sum. $$\sum_{i=1}^{45}(3 i-4)$$

Find the general antiderivative. $$\int \frac{x+2 x^{3 / 4}}{x^{5 / 4}} d x$$

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

Use the properties of logarithms to rewrite the expression as a single term. $$2 \ln 3-\ln 9+\ln \sqrt{3}$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{\pi / 2}^{\pi}(2 \sin x-\cos x) d x$$

Construct a table of Riemann sums as in example 3.5 to show that sums with right-endpoint, midpoint and Ieft-endpoint evaluation all converge to the same value as \(n \rightarrow \infty\) $$f(x)=4-x^{2},[-2,2]$$

Write the given (total) area as an integral or sum of integrals. The area above the \(x\) -axis and below \(y=4-x^{2}\).

Use summation rules to compute the sum. $$\sum_{i=1}^{40}\left(4-i^{2}\right)$$

Find the general antiderivative. $$\int(2 \sin x+\cos x) d x$$

Evaluate the indicated integral. $$\int e^{x} \sqrt{e^{x}+4} d x$$

Use the properties of logarithms to rewrite the expression as a single term. $$2 \ln \left(\frac{1}{3}\right)-\ln 3+\ln \left(\frac{1}{9}\right)$$

Use a computer or calculator to compute the Midpoint, Trapezoidal and Simpson's Rule approximations with \(n=10, n=20\) and \(n=50 .\) Compare these values to the approximation given by your calculator or computer. $$\int_{0}^{1} \sqrt[3]{x^{2}+1} d x$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{0}^{1}\left(e^{x}-e^{-x}\right) d x$$

Write the given (total) area as an integral or sum of integrals. The area above the \(x\) -axis and below \(y=4 x-x^{2}\).

Construct a table of Riemann sums as in example 3.5 to show that sums with right-endpoint, midpoint and Ieft-endpoint evaluation all converge to the same value as \(n \rightarrow \infty\) $$f(x)=\sin x,[0, \pi / 2]$$

Use summation rules to compute the sum. $$\sum_{i=1}^{50}(8-i)$$

Find the general antiderivative. $$\int(3 \cos x-\sin x) d x$$

Evaluate the indicated integral. $$\int \frac{e^{\sqrt{x}}}{\sqrt{x}} d x$$

Evaluate the derivative using properties of logarithms where needed. $$\frac{d}{d x}(\ln \sqrt{x^{2}+1})$$

Compute the exact value and compute the error (the difference between the approximation and the exact value) in each of the Midpoint, Trapezoidal and Simpson's Rule approximations using \(n=10, n=20, n=40\) and \(n=80\). $$\int_{0}^{1} 5 x^{4} d x$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{0}^{1 / 2} \frac{3}{\sqrt{1-x^{2}}} d x$$

Write the given (total) area as an integral or sum of integrals. The area below the \(x\) -axis and above \(y=x^{2}-4\).

Use summation rules to compute the sum. $$\sum_{i=1}^{100}\left(i^{2}-3 i+2\right)$$

Find the general antiderivative. $$\int 2 \sec x \tan x \, d x$$

Evaluate the indicated integral. $$\int \frac{x+1}{\left(x^{2}+2 x-1\right)^{2}} d x$$

Evaluate the derivative using properties of logarithms where needed. $$\frac{d}{d x}\left[\ln \left(x^{5} \sin x \cos x\right)\right]$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{-1}^{1} \frac{4}{1+x^{2}} d x$$

Write the given (total) area as an integral or sum of integrals. The area below the \(x\) -axis and above \(y=x^{2}-4 x\).

Construct a table of Riemann sums as in example 3.5 to show that sums with right-endpoint, midpoint and Ieft-endpoint evaluation all converge to the same value as \(n \rightarrow \infty\) $$f(x)=x^{3}-1,[-1,1]$$

Use summation rules to compute the sum. $$\sum_{i=1}^{140}\left(i^{2}+2 i-4\right)$$

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

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

Evaluate the derivative using properties of logarithms where needed. $$\frac{d}{d x}\left(\ln \frac{x^{4}}{x^{5}+1}\right)$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{1}^{4} \frac{x-3}{x} d x$$

Write the given (total) area as an integral or sum of integrals. The area between \(y=\sin x\) and the \(x\) -axis for \(0 \leq x \leq \pi\).

Use Riemann sums and a limit to compute the exact area under the curve. $$y=x^{2}+1 \text { on }[0,1]$$