91Ó°ÊÓ

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int x(x+1)^{2} d x $$

$$ \text { Evaluate the definite integral. } $$ $$ \int_{1}^{5} \frac{x-1}{x^{2}(x+1)} d x $$

Use the error formulas to find bounds for the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule. (Let \(n=4 .\).) $$ \int_{0}^{2} x^{3} d x $$

Use a spreadsheet to complete the table for the specified values of \(a\) and \(n\) to demonstrate that \(\lim _{x \rightarrow \infty} x^{n} e^{-a x}=0, \quad a>0, n>0\) \begin{tabular}{|l|l|l|l|l|} \hline\(x\) & 1 & 10 & 25 & 50 \\ \hline\(x^{n} e^{-a x}\) & & & & \\ \hline \end{tabular} $$ a=1, n=1 $$

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{x}{\sqrt{2+3 x}} d x $$

Use a spreadsheet to complete the table for the specified values of \(a\) and \(n\) to demonstrate that \(\lim _{x \rightarrow \infty} x^{n} e^{-a x}=0, \quad a>0, n>0\) \begin{tabular}{|l|l|l|l|l|} \hline\(x\) & 1 & 10 & 25 & 50 \\ \hline\(x^{n} e^{-a x}\) & & & & \\ \hline \end{tabular} $$ a=2, n=4 $$

$$ \text { Evaluate the definite integral. } $$ $$ \int_{0}^{1} \frac{x^{2}-x}{x^{2}+x+1} d x $$

Use the error formulas to find bounds for the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule. (Let \(n=4 .\).) $$ \int_{0}^{1} \frac{1}{x+1} d x $$

$$ \text { Evaluate the definite integral. } $$ $$ \int_{0}^{1} \frac{x^{3}}{x^{2}-2} d x $$

Use the error formulas to find bounds for the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule. (Let \(n=4 .\).) $$ \int_{0}^{1} e^{x^{3}} d x $$