91Ó°ÊÓ

Use a change of variables to find the following indefinite integrals. Check your work by differentiating. $$\int \frac{8 x+6}{2 x^{2}+3 x} d x$$

Find the following integrals. $$\int \frac{x}{\sqrt{x-4}} d x$$

Net area from graphs The figure shows the areas of regions bounded by the graph of \(f\) and the \(x\) -axis. Evaluate the following integrals. $$\int_{0}^{a} f(x) d x$$

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{1}^{9} \frac{2}{\sqrt{x}} d x$$

Complete the following steps for the given function, interval, and value of \(n\) a. Sketch the graph of the function on the given interval. b. Calculate \(\Delta x\) and the grid points \(x_{0}, x_{1}, \ldots, x_{n}\) c. Illustrate the midpoint Riemann sum by sketching the appropriate rectangles. d. Calculate the midpoint Riemann sum. $$f(x)=\frac{1}{x} \text { on }[1,6] ; n=5$$

Average height of an arch The height of an arch above the ground is given by the function \(y=10 \sin x,\) for \(0 \leq x \leq \pi\). What is the average height of the arch above the ground?

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{4}^{9} \frac{2+\sqrt{t}}{t} d t$$

Complete the following steps for the given function, interval, and value of \(n\) a. Sketch the graph of the function on the given interval. b. Calculate \(\Delta x\) and the grid points \(x_{0}, x_{1}, \ldots, x_{n}\) c. Illustrate the midpoint Riemann sum by sketching the appropriate rectangles. d. Calculate the midpoint Riemann sum. $$f(x)=4-x \text { on }[-1,4] ; n=5$$

Find the following integrals. $$\int \frac{y^{2}}{(y+1)^{4}} d y$$

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{-2}^{2}\left(x^{2}-4\right) d x$$