91Ó°ÊÓ

Find the integrals. Check your answers by differentiation. $$\int \frac{d x}{1+2 x^{2}}$$

Anti differentiate using the table of integrals. You may need to transform the integrals first. $$\int \frac{1}{1+(z+2)^{2}} d z$$

Complete the square and give a substitution (not necessarily trigonometric) which could be used to compute the integrals. $$\int \frac{d y}{y^{2}+3 y+3}$$

Calculate the integrals in Exercises \(5-33,\) if they converge. You may calculate the limits by appealing to the dominance of one function over another, or by l'Hopital's rule. $$\int_{3}^{6} \frac{d \theta}{(4-\theta)^{2}}$$

Evaluate the integrals, both exactly [e.g. \(\ln (3 \pi)] \text { and numerically [e.g. } \ln (3 \pi) \approx 2.243].\) $$\int_{3}^{5} x \cos x d x$$

Explain what is wrong with the statement. If \(0 \leq f(x) \leq g(x)\) and \(\int_{0}^{\infty} g(x) d x\) diverges then by the comparison test \(\int_{0}^{\infty} f(x) d x\) diverges.

Find the integrals. Check your answers by differentiation. $$\int \frac{d x}{\sqrt{1-4 x^{2}}}$$

Complete the square and give a substitution (not necessarily trigonometric) which could be used to compute the integrals. $$\int \frac{x+1}{x^{2}+2 x+2} d x$$

Complete the square and give a substitution (not necessarily trigonometric) which could be used to compute the integrals. $$\int \frac{4}{\sqrt{2 z-z^{2}}} d z$$

In statistics we encounter \(P(x)\), a function defined by $$P(x)=\frac{1}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} d t$$ Use a calculator or computer to evaluate (a) \(P(1)\) (b) \(P(\infty)\)