Exercises

Exercises 3.5 Exercises

1.

Use integration by parts to evaluate the following indefinite integrals.

\(\displaystyle \int 2x\arctan(5x) dx.\)
\(\displaystyle \int x^2e^{x/7}dx.\)
\(\displaystyle \int \ln(x^2 +23x + 60)dx.\)
\(\displaystyle \int x\cos^2(4x)dx.\)
\(\displaystyle \int x^2\sin(3x)dx.\)
\(\displaystyle \int \sqrt{x}\ln(x)dx.\)
\(\displaystyle \int -3x^4(\ln(x))^2dx.\)
\(\displaystyle \int 3x5^xdx.\)
\(\displaystyle \int x^3e^{x^2}dx.\)
\(\displaystyle \int 7\cos(\ln(x))dx.\)
\(\displaystyle \int -3x\sec^2(x)dx\)

2.

Evaluation the definite integral \(\int_0^{1/2} \arccos(x) dx\text{.}\)

3.

Derive the following reduction formulas.

\(\int x^ne^x dx = x^ne^x - n \int x^{n-1}e^x dx\text{.}\) Use this formula to evaluate \(\int x^3e^x dx\text{.}\)
For \(n \ge 2\text{,}\)
\begin{equation*} \int \cos^n(x)dx = \frac{1}{n}\cos^{n-1}(x)\sin(x) + \frac{n-1}{n}\int \cos^{n-2}(x) dx. \end{equation*}
One can deduce it from (3.2) using the substitution \(u=x + \frac{\pi}{2}.\)
For \(n \ge 2\text{,}\)
\begin{equation*} \int \tan^n(x)dx = \frac{1}{n-1}\tan^{n-1}(x) - \int \tan^{n-2}(x) dx. \end{equation*}

Hint.
\(\int \tan^n(x)dx = \int \tan^{n-2}(x)(\sec^2(x)-1) dx\)\(d\tan(x) = \sec^2(x)dx\text{.}\)
For \(n \ge 2\text{,}\)
\begin{equation*} \int \sec^n(x)dx = \frac{1}{n-1}\sec^{n-2}(x) \tan(x) + \frac{n-2}{n-1}\int\sec^{n-2}(x) dx. \end{equation*}

Hint.
\(\int\sec^n(x)dx = \int\sec^{n-2}(x)d\tan(x)\)

4.

Deduce from the reduction formula (3.2) that

\begin{equation*} \int_0^{\pi/2} \sin^n(x) dx = \frac{n-1}{n}\int_0^{\pi/2} \sin^{n-2}(x) dx. \end{equation*}

Conclude that

\begin{equation*} \int_0^{\pi/2} \sin^n(x)dx = \begin{cases} \frac{n-1}{n}\frac{n-3}{n-2}\cdots\frac{1}{2}\frac{\pi}{2} & n\ \text{even,} \\ \frac{n-1}{n}\frac{n-3}{n-2}\cdots\frac{2}{3} & n\ \text{odd.} \\ \end{cases} \end{equation*}

5.

Suppose \(f\) is infinitely differentiable. Show that for each \(n \ge 1\text{,}\)

\begin{equation*} \int e^x f(x) dx = e^x\sum_{k=0}^{n-1} (-1)^k f^{(k)}(x) + (-1)^n \int e^xf^{(n)}(x) dx. \end{equation*}

Use the above observation to compute the following integrals

\(\int x^2e^x dx\text{.}\)
\(\int e^x\sin(x) dx\text{.}\)

6.

Suppose \(f\) is infinitely differentiable. Show that for each \(n \ge 1\text{,}\)

\begin{equation*} \int e^{-x} f(x) dx = -e^{-x}\sum_{k=0}^{n-1} f^{(k)}(x) + \int e^{-x}f^{(n)}(x) dx. \end{equation*}

7.

Compute \(\displaystyle \int \frac{dx}{x\sqrt{x^2-1}}\) using substitution \(u = \sqrt{x^2-1}\) on both \(x \ge 1\) and \(x \lt -1\text{.}\) Verfiy the answer that you get differ from the one obtained in Example 3.29 by a constant on each interval.

8.

Evaluate the integral \(\displaystyle \int \frac{x\arctan(x)}{(1+x^2)^2}dx\text{.}\)

Solution.

The substitution \(x = \tan(\theta)\) turns the integral into \(\displaystyle \int \theta \sin(\theta)\cos(\theta) d\theta\text{.}\) Then an integration by parts with \(u=\theta\) and \(dv = \sin(\theta)\cos(\theta)d\theta\) turns the integral into

\begin{equation*} \frac{1}{2}\left(\theta \sin^2(\theta) - \int \sin^2(\theta) d\theta\right) \end{equation*}

Finally, working out the trigonometry integral \(\displaystyle \int \sin^2(\theta) d\theta\) and then expressing the answers back in terms of \(x\) yields

\begin{equation*} \frac{1}{2}\left( \frac{x^2\arctan(x)}{1+x^2} + \frac{x}{2(1+x^2)}-\frac{\arctan(x)}{2}\right) + C. \end{equation*}

9.

Let \(f\) be a function with \(f''\) continuous and \(f(0) = 7, f(1)=5\text{,}\) and \(f'(1)=7\text{.}\) Evaluate the integral \(\displaystyle \int_0^1 xf''(x)dx\text{.}\)

Solution.

\(u=x\)\(dv = f''(x)dx\text{,}\)\(du = dx\)\(v\)\(f'(x)\text{,}\)

\begin{gather*} \left. xf'(x) \right|_{0}^{1} - \int_0^1 f'(x)dx \end{gather*}

\begin{gather*} (1)f'(1) - 0f'(0) - (f(1)-f(0)) = f'(1)-f(1)+f(0) = 9 \end{gather*}

10.

Let \(I_s= \int e^{ax}\sin(bx) dx\) and \(I_c = \int e^{ax}\cos(bx)dx\) where \(a,b\) are two nonzero real numbers. Show that

\begin{align*} aI_s & = e^{ax}\sin(bx) - bI_c \quad \text{and}\\ aI_c & = e^{ax}\cos(bx) + bI_s. \end{align*}

Then use these equations to compute \(\int e^{2x}\cos(3x) dx\) and \(\int e^{2x}\sin(3x) dx\) (Hint: set \(a=2\) and \(b=3\))

Hint.

Use integration by parts on \(aI_s\) ( \(aI_c\)) to show the first (second) equality.

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