Mathematics

# Integrate with respect to $'x'$:$x\tan ^{ -1 }{ x }$

##### SOLUTION
$\displaystyle\int x{\text{tan}^{-1}x}dx=\text{tan}^{-1}x\displaystyle\int x{dx}-\displaystyle\int \dfrac{d(\text{tan}^{-1}x)}{dx}\bigg(\displaystyle\int x{dx}\bigg)dx$
$=\dfrac{x^{2}\text{tan}^{-1}x}{2}-\dfrac{1}{2}\displaystyle\int \dfrac{x^{2}+1-1}{1+x^{2}}dx\\=\dfrac{x^{2}\text{tan}^{-1}x}{2}-\dfrac{x}{2}+\dfrac{\text{tan}^{-1}x}{2}$
$=\dfrac{(x^{2}+1)\text{tan}^{-1}x-x}{2}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109

#### Realted Questions

Q1 Assertion & Reason Medium
##### ASSERTION

The value of $\int_{0}^{\pi / 4} \log (1+\tan \theta) d \theta=\frac{\pi}{8} \log 2$

##### REASON

The value of $\int_{0}^{\pi / 2} \log \sin \theta d \theta=-\pi \log 2$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Assertion is correct but Reason is incorrect
• C. Assertion is incorrect but Reason is correct
• D. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate $\displaystyle\int \sin^2(2x+5) dx$.

1 Verified Answer | Published on 17th 09, 2020

Q3 Assertion & Reason Hard
##### ASSERTION

The value of $\displaystyle \int_{0}^{\pi }xf\left ( \sin x \right )dx$ is $\displaystyle \frac{\pi }{2}\displaystyle \int_{0}^{\pi }f\left ( \sin x \right )dx$ or $\pi \displaystyle \int_{0}^{\pi /2}f\left ( \sin x \right )dx$

##### REASON

$\displaystyle \int_{0}^{a}f\left ( x \right )dx=\displaystyle \int_{0}^{a}f\left ( a-x \right )dx$ and $\displaystyle \int_{0}^{2a}f\left ( x \right )dx=2\displaystyle \int_{0}^{a}f\left ( x \right )dx$ If $f\left ( 2a-x \right )=f\left ( x \right )$

• A. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• B. Assertion is correct but Reason is incorrect
• C. Both Assertion and Reason are incorrect
• D. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle\int \cos \left[2\cot^{-1}\sqrt{\dfrac{1-x}{1+x}}\right]dx$ is equal to?
• A. $\dfrac{1}{2}\sin \left[2\cot^{-1}\sqrt{\dfrac{1-x}{1+x}}\right]+c$
• B. $-\dfrac{1}{2}x^2+c$
• C. $\dfrac{1}{2}x+c$
• D. $\dfrac{1}{2}x^2+c$

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard
Let us consider the integral of the following forms
$f{(x_1,\sqrt{mx^2+nx+p})}^{\tfrac{1}{2}}$
Case I If $m>0$, then put $\sqrt{mx^2+nx+C}=u\pm x\sqrt{m}$
Case II If $p>0$, then put $\sqrt{mx^2+nx+C}=u\pm \sqrt{p}$
Case III If quadratic equation $mx^2+nx+p=0$ has real roots $\alpha$ and $\beta$, then put $\sqrt{mx^2+nx+p}=(x-\alpha)u\:or\:(x-\beta)u$