Chapter 5

Evaluate the integrals. $$\int_{0}^{1 / 2} \frac{4}{\sqrt{1-x^{2}}} d x$$

Evaluate the integrals. $$\int \frac{\sec z \tan z}{\sqrt{\sec z}} d z$$

Evaluate the integrals. $$\int_{0}^{1 / \sqrt{3}} \frac{d x}{1+4 x^{2}}$$

Evaluate the sums. a. \(\sum_{k=1}^{n}\left(\frac{1}{n}+2 n\right)\) b. \(\sum_{k=1}^{n} \frac{c}{n}\) c. \(\sum_{k=1}^{n} \frac{k}{n^{2}}\)

Evaluate the integrals. $$\int \frac{1}{t^{2}} \cos \left(\frac{1}{t}-1\right) d t$$

Evaluate the integrals. $$\int_{2}^{4} x^{\pi-1} d x$$

Graph each function \(f(x)\) over the given interval. Partition the interval into four subintervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum \(\Sigma_{k=1}^{4} f\left(c_{k}\right) \Delta x_{k},\) given that \(c_{k}\) is the (a) left-hand endpoint, (b) righthand endpoint, (c) midpoint of the \(k\) th subinterval. (Make a separate sketch for each set of rectangles.) $$f(x)=x^{2}-1, \quad[0,2]$$

Evaluate the integrals. $$\int \frac{1}{\sqrt{t}} \cos (\sqrt{t}+3) d t$$

Evaluate the integrals. $$\int_{-1}^{0} \pi^{x-1} d x$$

Graph each function \(f(x)\) over the given interval. Partition the interval into four subintervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum \(\Sigma_{k=1}^{4} f\left(c_{k}\right) \Delta x_{k},\) given that \(c_{k}\) is the (a) left-hand endpoint, (b) righthand endpoint, (c) midpoint of the \(k\) th subinterval. (Make a separate sketch for each set of rectangles.) $$f(x)=-x^{2}, \quad[0,1]$$

Evaluate the integrals. $$\int \frac{1}{\theta^{2}} \sin \frac{1}{\theta} \cos \frac{1}{\theta} d \theta$$

Guess an antiderivative for the integrand function. Validate your guess by differentiation and then evaluate the given definite integral. (Hint: Keep in mind the Chain Rule in guessing an antiderivative. You will learn how to find such antiderivatives in the next section.) $$\int_{0}^{1} x e^{x^{2}} d x$$

Graph each function \(f(x)\) over the given interval. Partition the interval into four subintervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum \(\Sigma_{k=1}^{4} f\left(c_{k}\right) \Delta x_{k},\) given that \(c_{k}\) is the (a) left-hand endpoint, (b) righthand endpoint, (c) midpoint of the \(k\) th subinterval. (Make a separate sketch for each set of rectangles.) $$f(x)=\sin x, \quad[-\pi, \pi]$$

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-46\). $$\int_{0}^{\pi / 12} 6 \tan 3 x d x$$

Evaluate the integrals. $$\int \frac{\cos \sqrt{\theta}}{\sqrt{\theta} \sin ^{2} \sqrt{\theta}} d \theta$$

Guess an antiderivative for the integrand function. Validate your guess by differentiation and then evaluate the given definite integral. (Hint: Keep in mind the Chain Rule in guessing an antiderivative. You will learn how to find such antiderivatives in the next section.) $$\int_{1}^{2} \frac{\ln x}{x} d x$$

Graph each function \(f(x)\) over the given interval. Partition the interval into four subintervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum \(\Sigma_{k=1}^{4} f\left(c_{k}\right) \Delta x_{k},\) given that \(c_{k}\) is the (a) left-hand endpoint, (b) righthand endpoint, (c) midpoint of the \(k\) th subinterval. (Make a separate sketch for each set of rectangles.) $$f(x)=\sin x+1, \quad[-\pi, \pi]$$

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-46\). $$\int_{-\pi / 2}^{\pi / 2} \frac{2 \cos \theta d \theta}{1+(\sin \theta)^{2}}$$

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

Guess an antiderivative for the integrand function. Validate your guess by differentiation and then evaluate the given definite integral. (Hint: Keep in mind the Chain Rule in guessing an antiderivative. You will learn how to find such antiderivatives in the next section.) $$\int_{2}^{5} \frac{x d x}{\sqrt{1+x^{2}}}$$

Find the norm of the partition \(P=\\{0,1.2,1.5,2.3,2.6,3\\}\).

Evaluate the integrals. $$\int \sqrt{\frac{x-1}{x^{5}}} d x$$

Guess an antiderivative for the integrand function. Validate your guess by differentiation and then evaluate the given definite integral. (Hint: Keep in mind the Chain Rule in guessing an antiderivative. You will learn how to find such antiderivatives in the next section.) $$\int_{0}^{\pi / 3} \sin ^{2} x \cos x d x$$

Find the norm of the partition \(P=\\{-2,-1.6,-0.5,0,0.8,1\\}\).

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-46\). $$\int_{0}^{\ln \sqrt{3}} \frac{e^{x} d x}{1+e^{2 x}}$$

Evaluate the integrals. $$\int \frac{1}{x^{2}} \sqrt{2-\frac{1}{x}} d x$$

Find the derivatives. a. by evaluating the integral and differentiating the result. b. by differentiating the integral directly. $$\frac{d}{d x} \int_{0}^{\sqrt{x}} \cos t d t$$

Find a formula for the Riemann sum obtained by dividing the interval \([a, b]\) into \(n\) equal subintervals and using the right-hand endpoint for each \(c_{k} .\) Then take a limit of these sums as \(n \rightarrow \infty\) to calculate the area under the curve over \([a, b]\). \(f(x)=1-x^{2}\) over the interval [0,1]

Evaluate the integrals. $$\int \frac{1}{x^{3}} \sqrt{\frac{x^{2}-1}{x^{2}}} d x$$

Find the derivatives. a. by evaluating the integral and differentiating the result. b. by differentiating the integral directly. $$\frac{d}{d x} \int_{1}^{\sin x} 3 t^{2} d t$$

Find a formula for the Riemann sum obtained by dividing the interval \([a, b]\) into \(n\) equal subintervals and using the right-hand endpoint for each \(c_{k} .\) Then take a limit of these sums as \(n \rightarrow \infty\) to calculate the area under the curve over \([a, b]\). \(f(x)=2 x\) over the interval [0,3]

Evaluate the integrals. $$\int \sqrt{\frac{x^{3}-3}{x^{11}}} d x$$

Find the derivatives. a. by evaluating the integral and differentiating the result. b. by differentiating the integral directly. $$\frac{d}{d t} \int_{0}^{t^{4}} \sqrt{u} d u$$

Find a formula for the Riemann sum obtained by dividing the interval \([a, b]\) into \(n\) equal subintervals and using the right-hand endpoint for each \(c_{k} .\) Then take a limit of these sums as \(n \rightarrow \infty\) to calculate the area under the curve over \([a, b]\). \(f(x)=x^{2}+1\) over the interval [0,3]

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

Find the derivatives. a. by evaluating the integral and differentiating the result. b. by differentiating the integral directly. $$\frac{d}{d \theta} \int_{0}^{\tan \theta} \sec ^{2} y d y$$

Find a formula for the Riemann sum obtained by dividing the interval \([a, b]\) into \(n\) equal subintervals and using the right-hand endpoint for each \(c_{k} .\) Then take a limit of these sums as \(n \rightarrow \infty\) to calculate the area under the curve over \([a, b]\). \(f(x)=3 x^{2}\) over the interval [0,1]

Evaluate the integrals. $$\int x(x-1)^{10} d x$$

Find the derivatives. a. by evaluating the integral and differentiating the result. b. by differentiating the integral directly. $$\frac{d}{d x} \int_{0}^{x^{3}} e^{-t} d t$$

Find a formula for the Riemann sum obtained by dividing the interval \([a, b]\) into \(n\) equal subintervals and using the right-hand endpoint for each \(c_{k} .\) Then take a limit of these sums as \(n \rightarrow \infty\) to calculate the area under the curve over \([a, b]\). \(f(x)=x+x^{2}\) over the interval [0,1]

Evaluate the integrals. $$\int x \sqrt{4-x} d x$$

Find the derivatives. a. by evaluating the integral and differentiating the result. b. by differentiating the integral directly. $$\frac{d}{d t} \int_{0}^{\sqrt{t}}\left(x^{4}+\frac{3}{\sqrt{1-x^{2}}}\right) d x$$

Find a formula for the Riemann sum obtained by dividing the interval \([a, b]\) into \(n\) equal subintervals and using the right-hand endpoint for each \(c_{k} .\) Then take a limit of these sums as \(n \rightarrow \infty\) to calculate the area under the curve over \([a, b]\). \(f(x)=3 x+2 x^{2}\) over the interval [0,1]

Evaluate the integrals. $$\int(x+1)^{2}(1-x)^{5} d x$$

Find \(d y / d x\). $$y=\int_{0}^{x} \sqrt{1+t^{2}} d t$$

Find a formula for the Riemann sum obtained by dividing the interval \([a, b]\) into \(n\) equal subintervals and using the right-hand endpoint for each \(c_{k} .\) Then take a limit of these sums as \(n \rightarrow \infty\) to calculate the area under the curve over \([a, b]\). \(f(x)=2 x^{3}\) over the interval [0,1]

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-46\). $$\int_{0}^{3} \frac{y d y}{\sqrt{5 y+1}}$$

Evaluate the integrals. $$\int(x+5)(x-5)^{1 / 3} d x$$

Find \(d y / d x\). $$y=\int_{1}^{x} \frac{1}{t} d t, \quad x>0$$

Find a formula for the Riemann sum obtained by dividing the interval \([a, b]\) into \(n\) equal subintervals and using the right-hand endpoint for each \(c_{k} .\) Then take a limit of these sums as \(n \rightarrow \infty\) to calculate the area under the curve over \([a, b]\). \(f(x)=x^{2}-x^{3}\) over the interval [-1,0]