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. \(f(x)=4-x\) from \(x=0\) to \(x=4\)

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

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 \(x=1\)

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

Find the Gini index for the given Lorenz curve. Source: United Nations University-World Income Inequality Database \(L(x)=x^{15.3}\) (the Lorenz curve for wealth in Great Britain in 1990 )

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

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

Find each indefinite integral. \(\int\left(e^{2 x}-\frac{2}{x}\right) d x\)

Find each indefinite integral. \(\int\left(\frac{4}{z^{3}}+\frac{1}{\sqrt{z}}\right) d z\)

Find the Gini index for the given Lorenz curve. Source: United Nations University-World Income Inequality Database \(L(x)=0.4 x+0.6 x^{2}\)