Exam 7: Techniques of Integration

Write the form of the partial fraction decomposition of the rational expression. Do not find the numerical values of the constants. $\frac{x^{2}-x-8}{2 x^{3}-3 x^{2}+4 x-6}$

Find the integral. $\int \frac{d x}{\sqrt{x^{2}-4 x}}$

Find the integral. $\int x \tan ^{2} 3 x d x$

A particle moves on a straight line with velocity function $v(t)=\sin \omega t \cos ^{3} \omega t$ . Find its position function $s=f(t) \text { if } f(0)=0$ .

Evaluate the integral using the indicated trigonometric substitution. $\int \frac{x^{3}}{\sqrt{x^{2}+16}} d x ; x=4 \tan \theta$

Evaluate the integral. $\int 4 y^{3} e^{y} d y$

Estimate the area of the shaded region by using the Trapezoidal Rule with $n=4$ . Round the answer to the nearest tenth.

Evaluate the integral. $\int_{2}^{\infty} \frac{4 d y}{y^{2}+2 y-3}$

Find the integral. $\int \frac{3 x-5}{x^{2}-2 x-3} d x$

Find a bound on the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule with n subintervals. $\int_{-2}^{6} x^{3} d x ; \quad n=6$

Evaluate the integral. $\int e^{5 x} \cos 8 x d x$

Evaluate the integral. $\int_{4}^{\sqrt{17}} \sqrt{x^{2}-16} d x$

Evaluate the integral. $\int \frac{6 \sin 2 x}{1+\cos ^{4} x} d x$

Find the integral. $\int \tan ^{3} x \sec ^{5} x d x$

Evaluate the integral or show that it is divergent. $\int_{1}^{\infty} \frac{\ln x}{x^{6}} d x$

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places. $\int_{-1}^{1} \sqrt{x^{2}+1} d x ; \quad n=6$

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places. $\int_{0}^{\pi / 2} \sqrt{4+\sin ^{2} x} d x ; \quad n=6$

Evaluate the integral using an appropriate trigonometric substitution. $\int_{0}^{\sqrt{2}} \frac{x^{2}}{\sqrt{4-x^{2}}} d x$

Evaluate the integral. $\int \frac{6}{e^{3 x}-e^{x}} d x$

Evaluate the integral. $\int \frac{\cos x}{16+\sin ^{2} x} d x$