Mathematics

# Solve: $\displaystyle\int \dfrac{x^3}{\sqrt{x^4 + 1}}dx$

##### SOLUTION
Now,
$\displaystyle\int \dfrac{x^3}{\sqrt{x^4 + 1}}dx$
$=\dfrac{1}{4}\displaystyle\int \dfrac{4x^3}{\sqrt{x^4 + 1}}dx$
$=\dfrac{1}{4}\displaystyle\int \dfrac{d(x^4+1)}{\sqrt{x^4 + 1}}dx$
$=\dfrac{1}{4}.2\sqrt{x^4+1}+c$ [ Where $c$ is integrating constant]
$=\dfrac{1}{2}\sqrt{x^4+1}+c$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Hard
$\displaystyle \int x \frac{\ln (x + \sqrt{1 + x^{2}})}{\sqrt{1 + x^{2}}}\, dx$ equals to
• A. $\displaystyle \frac{x}{2} \ln^{2} (x + \sqrt{1 + x^{2}}) - \frac{x}{\sqrt{1 + x^{2}}} + c$
• B. $\displaystyle \frac{x}{2} \ln^{2} (x + \sqrt{1 + x^{2}}) + \frac{x}{\sqrt{1 + x^{2}}} + c$
• C. $\sqrt{1 + x^{2}} \ln (x + \sqrt{1 + x^{2}}) + x + c$
• D. $\sqrt{1 + x^{2}} \ln (x + \sqrt{1 + x^{2}}) - x + c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
Integrate : $\displaystyle \int\sqrt{ax+b}dx$
• A. $\dfrac {3(ax+b)^{3/2}}{2a}+C$
• B. $\dfrac {1}{2\sqrt{ax+b}}+C$
• C. None of these
• D. $\dfrac {2(ax+b)^{3/2}}{3a}+C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Solve : $\displaystyle2 \underset{0}{\overset{\pi / 2}{\int}} x \, sin \, x \, dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
The integral $\displaystyle \int{\frac{\sec^2 x}{\left(\sec x + \tan x \right)^{9/2}}}$ dx equals (for some arbitrary constant $k$)
• A. $\displaystyle \frac{-1}{\left(\sec x + \tan x \right)^{11/2}} \space \left\{ \frac{1}{11}-\frac{1}{7} \left(\sec x + \tan x \right)^2 \right\} + k$
• B. $\displaystyle \frac{1}{\left(\sec x + \tan x \right)^{11/2}} \space \left\{ \frac{1}{11}-\frac{1}{7} \left(\sec x + \tan x \right)^2 \right\} + k$
• C. $\displaystyle \frac{1}{\left(\sec x + \tan x \right)^{11/2}} \left\{ \frac{1}{11}+\frac{1}{7} \left(\sec x + \tan x \right)^2 \right\} + k$
• D. $\displaystyle \frac{-1}{\left(\sec x + \tan x \right)^{11/2}} \left\{ \frac{1}{11}+\frac{1}{7} \left(\sec x + \tan x \right)^2 \right\} + k$

$\int \frac{2x^{2}}{3x^{4}2x} dx$