91Ó°ÊÓ

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

Evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. $$ \int_{-1}^{1}\left(t^{3}-9 t\right) d t $$

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. $$ \int_{0}^{2} \frac{1}{\sqrt{1+x^{3}}} d x $$

Write the limit as a definite integral on the interval \([a, b]\), where \(c_{i}\) is any point in the \(i\) th subinterval. Limit $$ \lim _{\|\Delta\| \rightarrow 0} \sum_{i=1}^{n}\left(\frac{3}{c_{i}^{2}}\right) \Delta x_{i} $$ Interval $$ [1,3] $$

Complete the table. Original Integral $$\int x\left(x^{2}+3\right) d x$$

Find the indefinite integral and check the result by differentiation. $$ \int x^{2}\left(x^{3}+5\right)^{4} d x $$

Complete the table. Original Integral $$\int \frac{1}{2 x^{3}} d x$$

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. $$ \int_{0}^{1} \sqrt{x} \sqrt{1-x} d x $$

Find the indefinite integral and check the result by differentiation. $$ \int x^{2}\left(x^{3}-1\right)^{4} d x $$

Evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. $$ \int_{1}^{2}\left(\frac{3}{x^{2}}-1\right) d x $$