Skip to main content

Exercises 3.5 Exercises

1.

Use integration by parts to evaluate the following indefinite integrals.

(a)

\(\int 2x\arctan(5x) dx.\)

(c)

\(\int \ln(x^2 +23x + 60)dx.\)

(f)

\(\int \sqrt{x}\ln(x)dx.\)

(g)

\(\int -3x^4(\ln(x))^2dx.\)

2.

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

3.

Derive the following reduction formulas.

(a)

\(\int x^ne^x dx = x^ne^x - n \int x^{n-1}e^x dx\text{.}\) Use this formula to evaluate \(\int x^3e^x dx\text{.}\)

(b)

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 (3.3) using the substitution \(u = x + \frac{\pi}{2}\text{.}\)

(c)

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. \(\int \tan^n(x)dx = \int \tan^{n-2}(x)(\sec^2(x)-1) dx\) and \(d\tan(x) = \sec^2(x)dx\text{.}\)

(d)

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. \(\int\sec^n(x)dx = \int\sec^{n-2}(x)d\tan(x)\) then use integration by parts.

4.

Deduce from the reduction formula (3.3) 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.

(a)

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

(b)

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

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 \(x \lt -1\text{.}\) Verify the answer that you get differs from the one obtained in Example 3.30 by a constant on each interval.

8.

Evaluate the integral \(\displaystyle \int \frac{x\arctan(x)}{(1+x^2)^2}dx\text{.}\)
Solution.
Using \(x = \tan(\theta)\text{,}\) the integral becomes \(\displaystyle \int \theta \sin(\theta)\cos(\theta) d\theta\text{.}\) Applying integration by parts with \(u=\theta\) and \(dv = \sin(\theta)\cos(\theta)d\theta\text{,}\) we get
\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.
Using IBP with \(u=x\) and \(dv = f''(x)dx\text{,}\) then \(du = dx\) and \(v\) can be chosen as \(f'(x)\text{,}\) so the integral equals
\begin{equation*} \begin{aligned} \left. xf'(x) \right|_{0}^{1} - \int_0^1 f'(x)dx \end{aligned} \end{equation*}
By the FTC we have that equals
\begin{equation*} \begin{aligned} (1)f'(1) - 0f'(0) - (f(1)-f(0)) = f'(1)-f(1)+f(0) = 9 \end{aligned} \end{equation*}

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