91Ó°ÊÓ

Use a graphing utility to evaluate the integral. Graph the region whose area is given by the definite integral. $$ \int_{1}^{5} x^{2} \sqrt{x-1} d x $$

Prove or disprove that there is at least one straight line normal to the graph of \(y=\cosh x\) at a point \((a, \cosh a)\) and also normal to the graph of \(y=\sinh x\) at a point \((c, \sinh c)\). [At a point on a graph, the normal line is the perpendicular to the tangent at that point. Also, \(\cosh x=\left(e^{x}+e^{-x}\right) / 2\) and \(\left.\sinh x=\left(e^{x}-e^{-x}\right) / 2 .\right]\)

Prove that \(\frac{d}{d x}\left[\int_{u(x)}^{v(x)} f(t) d t\right]=f(v(x)) v^{\prime}(x)-f(u(x)) u^{\prime}(x)\).

Use a graphing utility to evaluate the integral. Graph the region whose area is given by the definite integral. $$ \int_{0}^{3}\left(\theta+\cos \frac{\theta}{6}\right) d \theta $$

Use a graphing utility to evaluate the integral. Graph the region whose area is given by the definite integral. $$ \int_{0}^{2}\left(e^{-2 x}+2\right) d x $$

Find the indefinite integral in two ways. Explain any difference in the forms of the answers. $$ \int(2 x-1)^{2} d x $$

Find the indefinite integral in two ways. Explain any difference in the forms of the answers. $$ \int \sin x \cos x d x $$

Evaluate the integral using the properties of even and odd functions as an aid. $$ \int_{-2}^{2} x^{2}\left(x^{2}+1\right) d x $$

Evaluate the integral using the properties of even and odd functions as an aid. $$ \int_{-\pi / 2}^{\pi / 2} \sin ^{2} x \cos x d x $$

Evaluate the integral using the properties of even and odd functions as an aid. $$ \int_{-2}^{2} x\left(x^{2}+1\right)^{3} d x $$