91Ó°ÊÓ

Determine the integrals by making appropriate substitutions. \(\int 3 x^{2} e^{\left(x^{3}-1\right)} d x\)

Evaluate the following definite integrals. \(\int_{0}^{3} \frac{x}{\sqrt{x+1}} d x\)

Evaluate the following integrals: \(\int \frac{x}{e^{x}} d x\)

Determine the integrals by making appropriate substitutions. \(\int 2 x e^{-x^{2}} d x\)

Evaluate the following integrals: \(\int x^{2} e^{x} d x\)

A continuous stream of income is produced at the rate of \(20 e^{1-.03 t}\) thousand dollars per year at. time \(t\), and invested money earns \(6 \%\) interest. (a) Write a definite integral that gives the present value of this stream of income over the time from \(t=2\) to \(t=5\) years. (b) Compute the present value described in part (a).

Approximate the following integrals by the midpoint rule; then, find the exact value by integration. Express your answers to five decimal places. \(\int_{0}^{4}\left(x^{2}+5\right) d x ; n=2,4\)

A growth company is one whose net earnings tend to increase each year. Suppose that the net earnings of a company at time \(t\) are being generated at the rate of \(30+5 t\) million dollars per year. (a) Write a definite integral that gives the present value of the company's earnings over the next 2 years using a \(10 \%\) interest rate. (b) Compute the present value described in part (a).

Evaluate the following integrals: \(\int \frac{x}{\sqrt{x+1}} d x\)

Determine the integrals by making appropriate substitutions. \(\int x \sqrt{4-x^{2}} d x\)