009B Sample Midterm 3, Problem 5

Evaluate the indefinite and definite integrals.

(a)   $\int x\ln x~dx$

(b)   $\int _{0}^{\pi }\sin ^{2}x~dx$

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Solution

Detailed Solution

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Foundations:
1. Recall the trig identity
$\tan ^{2}x+1=\sec ^{2}x$
2. Recall the trig identity
$\sin ^{2}(x)={\frac {1-\cos(2x)}{2}}$
3. How would you integrate  $\tan x~dx?$
You could use  $u$-substitution.
First, write  $\int \tan x~dx=\int {\frac {\sin x}{\cos x}}~dx.$
Now, let  $u=\cos(x).$  Then,  $du=-\sin(x)dx.$
Thus,
${\begin{array}{rcl}\displaystyle {\int \tan x~dx}&=&\displaystyle {\int {\frac {-1}{u}}~du}\\&&\\&=&\displaystyle {-\ln(u)+C}\\&&\\&=&\displaystyle {-\ln |\cos x|+C.}\\\end{array}}$

Solution:

(a)

Step 1:
We start by writing
$\int \tan ^{3}x~dx=\int \tan ^{2}x\tan x~dx.$
Since  $\tan ^{2}x=\sec ^{2}x-1,$  we have
${\begin{array}{rcl}\displaystyle {\int \tan ^{3}x~dx}&=&\displaystyle {\int (\sec ^{2}x-1)\tan x~dx}\\&&\\&=&\displaystyle {\int \sec ^{2}x\tan x~dx-\int \tan x~dx.}\\\end{array}}$

Step 2:
Now, we need to use  $u$-substitution for the first integral.
Let  $u=\tan(x).$
Then,  $du=\sec ^{2}x~dx.$
So, we have
${\begin{array}{rcl}\displaystyle {\int \tan ^{3}x~dx}&=&\displaystyle {\int u~du-\int \tan x~dx}\\&&\\&=&\displaystyle {{\frac {u^{2}}{2}}-\int \tan x~dx}\\&&\\&=&\displaystyle {{\frac {\tan ^{2}x}{2}}-\int \tan x~dx.}\\\end{array}}$

Step 3:
For the remaining integral, we also need to use  $u$-substitution.
First, we write
$\int \tan ^{3}x~dx={\frac {\tan ^{2}x}{2}}-\int {\frac {\sin x}{\cos x}}~dx.$
Now, we let  $u=\cos x.$
Then,  $du=-\sin x~dx.$
Therefore, we get
${\begin{array}{rcl}\displaystyle {\int \tan ^{3}x~dx}&=&\displaystyle {{\frac {\tan ^{2}x}{2}}+\int {\frac {1}{u}}~dx}\\&&\\&=&\displaystyle {{\frac {\tan ^{2}x}{2}}+\ln |u|+C}\\&&\\&=&\displaystyle {{\frac {\tan ^{2}x}{2}}+\ln |\cos x|+C.}\end{array}}$

(b)

Step 1:
One of the double angle formulas is
$\cos(2x)=1-2\sin ^{2}(x).$
Solving for  $\sin ^{2}(x),$  we get
$\sin ^{2}(x)={\frac {1-\cos(2x)}{2}}.$
Plugging this identity into our integral, we get
${\begin{array}{rcl}\displaystyle {\int _{0}^{\pi }\sin ^{2}x~dx}&=&\displaystyle {\int _{0}^{\pi }{\frac {1-\cos(2x)}{2}}~dx}\\&&\\&=&\displaystyle {\int _{0}^{\pi }{\frac {1}{2}}~dx-\int _{0}^{\pi }{\frac {\cos(2x)}{2}}~dx.}\\\end{array}}$

Step 2:
If we integrate the first integral, we get
${\begin{array}{rcl}\displaystyle {\int _{0}^{\pi }\sin ^{2}x~dx}&=&\displaystyle {\left.{\frac {x}{2}}\right|_{0}^{\pi }-\int _{0}^{\pi }{\frac {\cos(2x)}{2}}~dx}\\&&\\&=&\displaystyle {{\frac {\pi }{2}}-\int _{0}^{\pi }{\frac {\cos(2x)}{2}}~dx.}\\\end{array}}$

Step 3:
For the remaining integral, we need to use  $u$-substitution.
Let  $u=2x.$
Then,  $du=2~dx$  and  ${\frac {du}{2}}=dx.$
Also, since this is a definite integral and we are using $u$-substitution,
we need to change the bounds of integration.
We have  $u_{1}=2(0)=0$  and  $u_{2}=2(\pi )=2\pi .$
So, the integral becomes
${\begin{array}{rcl}\displaystyle {\int _{0}^{\pi }\sin ^{2}x~dx}&=&\displaystyle {{\frac {\pi }{2}}-\int _{0}^{2\pi }{\frac {\cos(u)}{4}}~du}\\&&\\&=&\displaystyle {{\frac {\pi }{2}}-\left.{\frac {\sin(u)}{4}}\right|_{0}^{2\pi }}\\&&\\&=&\displaystyle {{\frac {\pi }{2}}-{\bigg (}{\frac {\sin(2\pi )}{4}}-{\frac {\sin(0)}{4}}{\bigg )}}\\&&\\&=&\displaystyle {{\frac {\pi }{2}}.}\\\end{array}}$

Final Answer:
(a)     ${\frac {\tan ^{2}x}{2}}+\ln |\cos x|+C$
(b)     ${\frac {\pi }{2}}$

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