91Ó°ÊÓ

Use a definite integral to find the area under each curve between the given \(x\) -values. For Exercises \(19-24\), also make a sketch of the curve showing the region. \(f(x)=x\) from \(x=0\) to \(x=4\)

Find each indefinite integral. $$ \int\left(10 \sqrt[3]{t^{2}}+\frac{1}{\sqrt[3]{t^{2}}}\right) d t $$

Find the average value of each function over the given interval. $$ f(x)=x^{n} \text { on }[0,1], \text { where } n \text { is a constant }(n>0) $$

Use a definite integral to find the area under each curve between the given \(x\) -values. For Exercises \(19-24\), also make a sketch of the curve showing the region. \(f(x)=4-x\) from \(x=0\) to \(x=4\)

Find each indefinite integral. \(\int\left(e^{3 x}-\frac{3}{x}\right) d x\)

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int \frac{d x}{1+5 x} $$

Find the average value of each function over the given interval. $$ f(x)=e^{k x} \text { on }[0,1], \text { where } k \text { is a constant }(k \neq 0) $$

Find each indefinite integral. $$ \int\left(21 \sqrt{t^{3}}+\frac{6}{\sqrt{t^{5}}}\right) d t $$

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int \frac{d x}{1+3 x} $$

Use a definite integral to find the area under each curve between the given \(x\) -values. For Exercises \(19-24\), also make a sketch of the curve showing the region. \(f(x)=1-x^{2}\) from \(x=-1\) to \(x=1\)