91Ó°ÊÓ

Estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than \(10^{-4}\) by (a) the Trapezoidal Rule and (b) Simpson's Rule. (The integrals in Exercises \(11-18 \text { are the integrals from Exercises } 1-8 .)\) $$\int_{0}^{2} \sin (x+1) d x$$

Evaluate the integrals using integration by parts. $$\int e^{\theta} \sin \theta d \theta$$

Use the table of integrals at the back of the book to evaluate the integrals. $$\int \sin 3 x \cos 2 x d x$$

Evaluate the integrals without using tables. $$\int_{-\infty}^{0} \theta e^{\theta} d \theta$$

Let \(f(x)=\frac{c}{1+x^{2}} .\) Find the value of \(c\) so that \(f\) is a probability density function. If \(f\) is a probability density function for the random variable \(X,\) find the probability \(P(1 \leq X<2)\)

Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. $$\int \frac{x+2 \sqrt{x-1}}{2 x \sqrt{x-1}} d x$$

Use any method to evaluate the integrals. Most will require trigonometric substitutions, but some can be evaluated by other methods. \(\int x \sqrt{x^{2}-4} d x\)

Evaluate the integrals. $$\int_{0}^{\pi / 2} \sin ^{2} 2 \theta \cos ^{3} 2 \theta d \theta$$

Evaluate the integrals without using tables. $$\int_{0}^{\infty} 2 e^{-\theta} \sin \theta d \theta$$

Express the integrand as a sum of partial fractions and evaluate the integrals. $$\int_{1}^{\sqrt{3}} \frac{3 t^{2}+t+4}{t^{3}+t} d t$$