91Ó°ÊÓ

Evaluate the definite integral. Use a graphing utility to verify your result. \(\int_{-1}^{1}(\sqrt[3]{t}-2) d t\)

In Exercises 11–32, find the indefinite integral and check the result by differentiation. $$ \int \sqrt[3]{x^{2}} d x $$

Finding an Indefinite Integral In Exercises \(5-26\) , find the indefinite integral and check the result by differentiation. $$ \int \frac{x^{3}}{\left(1+x^{4}\right)^{2}} d x $$

Using the Trapezoidal Rule and Simpson's Rule In Exercises \(11-20\) , approximate the definite integral using the Trapezoidal Rule and Simpson's Rule with \(n=4\) . Compare these results with the approximation of the integral using a graphing utility. $$ \int_{0}^{\pi / 2} \sqrt{1+\sin ^{2} x} d x $$

Evaluate the definite integral. Use a graphing utility to verify your result. \(\int_{1}^{8} \sqrt{\frac{2}{x}} d x\)

Evaluating a Sum In Exercises \(13-20\) , use the properties of summation and Theorem 4.2 to evaluate the sum. Use the summation capabilities of a graphing utility to verify your result. $$ \sum_{i=1}^{10}\left(i^{2}-1\right) $$

In Exercises 11–32, find the indefinite integral and check the result by differentiation. $$ \int\left(\sqrt[4]{x^{3}}+1\right) d x $$

Finding an Indefinite Integral In Exercises \(5-26\) , find the indefinite integral and check the result by differentiation. $$ \int \frac{x^{2}}{\left(1+x^{3}\right)^{2}} d x $$

Evaluating a Sum In Exercises \(13-20\) , use the properties of summation and Theorem 4.2 to evaluate the sum. Use the summation capabilities of a graphing utility to verify your result. $$ \sum_{i=1}^{15} i(i-1)^{2} $$

Using the Trapezoidal Rule and Simpson's Rule In Exercises \(11-20\) , approximate the definite integral using the Trapezoidal Rule and Simpson's Rule with \(n=4\) . Compare these results with the approximation of the integral using a graphing utility. $$ \int_{0}^{\pi / 4} x \tan x d x $$