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

Find the trigonometric integral. ∫ cos³(πx - 1) dx

Use the Special Integration Formulas to find the indefinite integral. ∫ √(25 - 4x²) dx

Find the indefinite integral. ∫ [v + 1/(3v - 1)³] dv

In this exercise, determine whether the improper integral diverges or converges. Evaluate the integral if it converges.

$$\displaystyle\int_0^{\infty} x e^{-x / 2} d x$$

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.

$$\int_0^4 \sqrt{x} e^x d x$$

Use Wallis’s Formulas to evaluate the integral. $\int_{0}^{\pi / 2} \sin ^{10} x dx$

Find the indefinite integral.

$$\int\left[z^2+\frac{1}{(1-z)^6}\right] d z$$

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. $\int_{0}^{4} \sqrt{x} e^{x} dx$

Use Wallis's Formulas to evaluate the integral.

$$\int_0^{\pi / 2} \sin ^{10} x d x$$

Evaluate the limit using L'Hopital's Rule i necessary. (In the previous Exercise, $n$ is a positive integer.)
$\lim _{x \rightarrow 0} \dfrac{\sin 2 x}{\sin 3 x}$

Find the indefinite integral. $\int\left[z^{2}+\frac{1}{(1-z)^{6}}\right] dz$

Find the indefinite integral.

$$\int \sqrt{16-4 x^2} d x$$

In this exercise, use Wallis's Formulas to evaluate the integral.

$$\displaystyle\int_0^{\pi / 2} \cos ^3 x d x$$