91Ó°ÊÓ

Show that the average value of \(f(x)\) over an interval \([a, b]\) is the same as the average value of \(f(c x)\) over the interval \(\left[\frac{a}{c}, \frac{b}{c}\right]\) for \(c>0\)

Find the area under the graph of \(f(t)=\frac{t}{\left(1+t^{2}\right)^{a}}\) between \(t=0\) and \(t=x\) where \(a>0\) and \(a \neq 1\) is fixed, and evaluate the limit as \(x \rightarrow \infty\) .

Find the area under the graph of \(g(t)=\frac{t}{\left(1-t^{2}\right)^{a}}\) between \(t=0\) and \(t=x,\) where \(0 < x < 1\) and \(a > 0\) is fixed. Evaluate the limit as \(x \rightarrow 1\)

The area of a semicircle of radius 1 can be expressed as \(\int_{-1}^{1} \sqrt{1-x^{2}} d x .\) Use the substitution \(x=\cos t\) to express the area of a semicircle as the integral of a trigonometric function. You do not need to compute the integral.

The area of the top half of an ellipse with a major axis that is the \(x\) -axis from \(x=-1\) to \(a\) and with a minor axis that is the \(y\) -axis from \(y=-b\) to \(b\) can be written as \(\int_{-a}^{a} b \sqrt{1-\frac{x^{2}}{a^{2}}} d x .\) Use the substitution \(x=a \cos t\) to express this area in terms of an integral of a trigonometric function. You do not need to compute the integral.

In the following exercises, compute each indefinite integral. $$\int e^{2 x} d x$$

In the following exercises, compute each indefinite integral. $$\int e^{-3 x} d x$$

In the following exercises, compute each indefinite integral. $$\int 2^{x} d x$$

In the following exercises, compute each indefinite integral. $$\int 3^{-x} d x$$

In the following exercises, compute each indefinite integral. $$\int \frac{1}{2 x} d x$$