Problem 78
Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals. $$\int \frac{d y}{y(\sqrt{a}-\sqrt{y})}, a>0(\text {Hint: Let } u=\sqrt{y}.)$$
Problem 79
Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals. $$\int \frac{\sec t}{1+\sin t} d t$$
Problem 79
Another Simpson's Rule formula Another Simpson's Rule formula is \(S(2 n)=\frac{2 M(n)+T(n)}{3},\) for \(n \geq 1 .\) Use this rule to estimate \(\int_{1}^{e} \frac{1}{x} d x\) using \(n=10\) subintervals.
Problem 79
Care with the secant substitution Recall that the substitution \(x=a \sec \theta\) implies either \(x \geq a\) (in which case \(0 \leq \theta < \pi / 2\) and \(\tan \theta \geq 0 \text { ) or } x \leq-a \text { (in which case } \pi / 2 < \theta \leq \pi \text { and } \tan \theta \leq 0)\). $$\text { Show that } \int \frac{d x}{x \sqrt{x^{2}-1}}=\left\\{\begin{array}{ll} \sec ^{-1} x+C & \text { if } x > 1 \\ -\sec ^{-1} x+C & \text { if } x < -1 \end{array}\right.$$
Problem 79
The following integrals may require more than one table look-up. Evaluate the integrals using a table of integrals, and then check your answer with a computer algebra system. $$\int x \sin ^{-1} 2 x d x$$
Problem 79
Are length Find the length of the curve \(y=x^{5 / 4}\) on the interval \([0,1] .\) (Hint: Write the are length integral and let \(u^{2}=1+\left(\frac{5}{4}\right)^{2} \sqrt{x}\)
Problem 79
Determine whether the following integrals converge or diverge. $$\int_{3}^{\infty} \frac{d x}{\ln x}(\text { Hint: } \ln x \leq x .)$$
Problem 80
An identity Show that if \(f\) and \(g\) have continuous second derivatives and \(f(0)=f(1)=g(0)=g(1)=0,\) then $$\int_{0}^{1} f^{\prime \prime}(x) g(x) d x=\int_{0}^{1} f(x) g^{\prime \prime}(x) d x.$$
Problem 80
Care with the secant substitution Recall that the substitution \(x=a \sec \theta\) implies either \(x \geq a\) (in which case \(0 \leq \theta < \pi / 2\) and \(\tan \theta \geq 0 \text { ) or } x \leq-a \text { (in which case } \pi / 2 < \theta \leq \pi \text { and } \tan \theta \leq 0)\). Evaluate for \(\int \frac{\sqrt{x^{2}-1}}{x^{3}} d x,\) for \(x > 1\) and for \(x < -1\)
Problem 80
The following integrals may require more than one table look-up. Evaluate the integrals using a table of integrals, and then check your answer with a computer algebra system. $$\int 4 x \cos ^{-1} 10 x d x$$
