91Ó°ÊÓ

Decide if the improper integral \(\int_{0}^{\infty} e^{-2 t} d t\) converges, and if so, to what value, by the following method. (a) Evaluate \(\int_{0}^{b} e^{-2 t} d t\) for \(b=3,5,7,10\). What do you observe? Make a guess about the convergence of the improper integral. (b) Find \(\int_{0}^{b} e^{-2 t} d t\) using the Fundamental Theorem. Your answer will contain \(b\). (c) Take a limit as \(b \rightarrow \infty\). Does your answer confirm your guess?

Find the indefinite integrals. $$ \int\left(t^{2}+5 t+1\right) d t $$

Find the integrals in problems. Check your answers by differentiation. $$ \int \frac{x+1}{x^{2}+2 x+19} d x $$

Find the indefinite integrals. $$ \int 5 e^{z} d z $$

(a) Evaluate \(\int_{0}^{b} x e^{-x / 10} d x\) for \(b=10,50,100,200\). (b) Assuming that it converges, estimate the value of \(\int_{0}^{\infty} x e^{-x / 10} d x\).

Find the integrals in problems. Check your answers by differentiation. $$ \int \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} d x $$

If appropriate, evaluate the following integrals by substitution. If substitution is not appropriate, say so, and do not evaluate. (a) \(\int x \sin \left(x^{2}\right) d x\) (b) \(\int x^{2} \sin x d x\) (c) \(\int \frac{x^{2}}{1+x^{2}} d x\) (d) \(\int \frac{x}{\left(1+x^{2}\right)^{2}} d x\) (e) \(\int x^{3} e^{x^{2}} d x\) (f) \(\int \frac{\sin x}{2+\cos x} d x\)

Find the indefinite integrals. $$ \int\left(t^{3}+6 t^{2}\right) d t $$

(a) Find \(\int(x+5)^{2} d x\) in two ways: (i) By multiplying out (ii) By substituting \(w=x+5\) (b) Are the results the same? Explain.

Find the exact area below the curve \(y=x^{3}(1-x)\) and above the \(x\) -axis.