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.

$$\displaystyle\int_{0}^{\pi/4}x \tan x \ dx$$

Determine which of the integrals can be found using the basic integration formulas you have studied so far in the text.

$$\displaystyle\int \frac{x^3}{1+x^4}\ dx$$

Find the derivative of the function.

$$y= \ln|\frac{\cos x}{1- \sin x}|$$

Let $f(x)=\sin(\ln x)$.

Determine the domain of the function $f$.

Determine which of the integrals can be found using the basic integration formulas you have studied so far in the text.

$$\displaystyle\int \frac{x}{1+x^4}\ dx$$

Think About It Use a graphing utility to graph $f(x)=\sin x$ and $g(x)=\arcsin (\sin x)$.

Determine the extrema of $g$.

Think About It Use a graphing utility to graph $f(x)=\sin x\ \text{and}\ g(x)= \arcsin (\sin x)$.

Find the indefinite integral.

$$\displaystyle\int (1-\sec\pi x)^2 \sec \pi x\tan \pi x\ dx$$

Verify each rule by differentiating (a>0).

$$\int \frac{d u}{u \sqrt{u^2-a^2}}=\frac{1}{a} \operatorname{arcsec} \frac{|u|}{a}+C$$

**Determine which of the integrals can be found using the basic integration formulas you have studied **

$$\displaystyle\int \frac{1}{1+x^4}\ dx$$

Verify each rule by differentiating (a>0).

$$\int \frac{d u}{a^2+u^2}=\frac{1}{a} \arctan \frac{u}{a}+C$$

Verify each rule by differentiating (a>0).

$$\int \frac{d u}{\sqrt{a^2-u^2}}=\arcsin \frac{u}{a}+C$$

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.

$$\displaystyle\int_{0}^{\pi/2} \sqrt{1+\sin^2 x} \ \ dx$$

Determine the limit of the trigonometric function (if it exists).

$$lim_{x\to0} \ \frac{x+2}{\cot x}$$