91Ó°ÊÓ

Use a substitution to change the integral into one you can find in the table. Then evaluate the integral. $$\int \frac{x^{2}+6 x}{\left(x^{2}+3\right)^{2}} d x$$

Use a substitution to change the integral into one you can find in the table. Then evaluate the integral. $$\int \sin ^{-1} \sqrt{x} d x$$

Evaluate the integrals without using tables. $$\int_{1}^{2} \frac{d s}{s \sqrt{s^{2}-1}}$$

Use any method to evaluate the integrals. Most will require trigonometric substitutions, but some can be evaluated by other methods. \(\int \frac{8 d x}{\left(4 x^{2}+1\right)^{2}}\)

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. $$\int(\csc x-\sec x)(\sin x+\cos x) d x$$

Prove that the sum \(T\) in the Trapezoidal Rule for \(\int_{a}^{b} f(x) d x\) is a Riemann sum for \(f\) continuous on \([a, b] .\) (Hint: Use the Intermediate Value Theorem to show the existence of \(c_{k}\) in the subinterval \(\left[x_{k-1}, x_{k}\right]\) satisfying \(f\left(c_{k}\right)=\left(f\left(x_{k-1}\right)+f\left(x_{k}\right)\right) / 2 .\) )

The digestion time in hours of a fixed amount of food is exponentially distributed with a mean of 1 hour. What is the probability that the food is digested in less than 30 minutes?

Evaluate the integrals. $$\int_{5 \pi / 6}^{\pi} \frac{\cos ^{4} x}{\sqrt{1-\sin x}} d x$$

Express the integrand as a sum of partial fractions and evaluate the integrals. $$\int \frac{x^{2}}{x^{4}-1} d x$$

Evaluate the integrals by using a substitution prior to integration by parts. $$\int \sin (\ln x) d x$$