Which, if any, of the integrals listed below can be found using the 20 basic integration rules? For any that can be found, do so. For any that cannot, explain why not.

$$\int \frac{3 x}{\sqrt{1-x^2}} d x$$

Evaluate the integrals ∫_-1^1 (1-x²)dx and ∫_-1^1 (1-x²)² dx.

Prove the following generalization of the Mean Value Theorem. If f is twice differentiable on the closed interval [a, b], then f(b) - f(a) = f′(a)(b - a) - ∫_a^b f″(t)(t - b) dt.

Explain why the evaluation of the integral is incorrect. Use the integration capabilities of a graphing utility to attempt to evaluate the integral.

$$∫_{-1}^1 (1/x²) dx = -2$$

Describe how you would find each integral using (i) integration by parts and (ii) substitution. Perform each integration. Are the results equivalent?

$$\int 2 x \sqrt{2 x-3} d x$$

Select the correct antiderivative.

$$\dfrac{d y}{d x}=\dfrac{x}{\sqrt{x^2+1}}$$

$$2 \sqrt{x^2+1}+C$$

Which, if any, of the integrals listed below can be found using the 20 basic integration rules? For any that can be found, do so. For any that cannot, explain why not.

$$\int \frac{3}{\sqrt{1-x^2}} d x$$

Consider the shaded region between the graph of $y=\sin x$, where $0 \leq x \leq \pi$, and the line $y=c$, where $0 \leq c \leq 1$, as shown in the figure. A solid is formed by revolving the region about the line $y=c$. (a) For what value of $c$ does the solid have minimum volume? (b) For what value of $c$ does the solid have maximum volume?

Power lines are constructed by stringing wire between supports and adjusting the tension on each span. The wire hangs between supports in the shape of a catenary, as shown in the figure. Let T be the tension (in pounds) on a span of wire, let u be the density (in pounds per foot), let g $\approx$ 32.2 be the acceleration due to gravity (in feet per second per second), and let L be the distance (in feet) between the supports. Then the equation of the catenary is $y=\frac{T}{u g}\left(\cosh \frac{u g x}{T}-1\right)$, where x and y are measured in feet. Find the length of the wire between two spans.

Find the indefinite integral. ∫ (ln x)² / x dx

In this exercise, use integration tables to find the integral.

$$\displaystyle\int x \operatorname{arcsec}\left(x^2+1\right) d x$$

Evaluate the integrals

$$\int_{-1}^1\left(1-x^2\right) d x \text { and } \int_{-1}^1\left(1-x^2\right)^2 d x .$$

Consider the function $f(x)=\sin ^4 x+\cos ^4 x$.

Use the power-reducing formulas to write the function in terms of the first power of the cosine.

Approximate the definite integral using the Trapezoidal Rule and Simpson’s Rule with n=4. Compare these results with the approximation of the integral using a graphing utility. $\int_{0}^{4} \sqrt{x} e^{x} d x$