Let $I_{m,n} = \int_{0}^{1}x^{m}(1-x)^{n}\ dx$ for constant $m,n.$ Show that $I_{m,n} = I_{n,m}.$

Evaluate the definite integrals using the Fundamental Theorem of Calculus and check your answers numerically.

$$\displaystyle \int_{-2}^{0} \frac{2x+4}{x^{2}+4x+5}\ dx$$

Evaluate the definite integrals using the Fundamental Theorem of Calculus and check your answers numerically.

$$\displaystyle \int_{-\pi/4}^{\pi/4}\cos^{2}\theta \sin^{5}\theta\ {d\theta}$$

Find a substitution $w$ and constants $a, b, k$ so that the integral has the form $\int_{a}^{b}kf(w)\ dw.$

$$\displaystyle \int_{1}^{9}f(6x\sqrt{x})\sqrt{x}\ dx$$

Evaluate the definite integrals using the Fundamental Theorem of Calculus and check your answers numerically.

$$\displaystyle \int_{0}^{1} \frac{dx}{x^{2}+1}$$

Compute approximations to $\int_{2}^{3}(1/x^{2})\ dx$

$$\text{TRAP}(2)$$

Evaluate the definite integrals using the Fundamental Theorem of Calculus and check your answers numerically.

$$\displaystyle \int_{-\pi/3}^{\pi/4} \sin^{3}\theta\cos\theta\ d\theta$$

Based on the previous exercise, you should now have three different expressions for the indefinite integral $\int \sin \theta \cos \theta d \theta$. Are they really different? Are they all correct? Explain.

There is yet another way of finding this integral which involves the trigonometric identities

$$\begin{aligned} \sin (2 \theta) & =2 \sin \theta \cos \theta \\ \cos (2 \theta) & =\cos ^2 \theta-\sin ^2 \theta . \end{aligned}$$

Find $\int \sin \theta \cos \theta d \theta$ using one of these identities and then the substitution $w=2 \theta$.

You probably solved the previous exercise by making the substitution w = sin $\theta$ or $w=\cos \theta$. (If not, go back and do it that way.) Now find $\int \sin \theta \cos \theta d \theta$ by making the other substitution.

Find the integral. Check your answers by differentiation.

$$\displaystyle \int y(y^{2}+5)^{8}\ dy$$

Find $\int \sin \theta \cos \theta d \theta$.

Evaluate the following integrals. Assume $a, b, c,$ and $k$ are constants. This exercise can be done without the integral table.

$$\displaystyle \int (x^{3} − 1)^{4}x^{2}\ dx$$

Explain how the expressions from the previous exercises are different. Are they both correct?