Mathematics

# $\int\dfrac{x-1}{(x+1)\sqrt{x^{3}+x^{2}+x^{}}}dx$ is equal to

$\tan^{-1} \sqrt{\dfrac{x^{2}+x+1}{x}}+c$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 114

#### Realted Questions

Q1 Subjective Hard
Evaluate the following integral:
$\int { \cfrac { { x }^{ 2 }+x+1 }{ { x }^{ 2 }-x } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
if $\displaystyle f(x)=\frac{e^x}{1+e^x},I_1=\int_{ f(-a) }^{f(a)}xg(x(1-x))\:dx$, and $\displaystyle I_2=\int_{ f(-a) }^{f(a)}g(x(1-x))\:dx$,then the value of $\dfrac{I_2}{I_1}$ is
• A. $-1$
• B. $-2$
• C. $1$
• D. $2$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\displaystyle \frac{2x^{3}+1}{(x-1)(x+1)(x+2)}=$
• A. $2-\dfrac{1}{2(x-1)}-\dfrac{1}{2(x+1)}-\dfrac{15}{x+2}$
• B. $2+\dfrac{1}{2(x-1)}-\dfrac{1}{2(x+1)}+\dfrac{15}{x+2}$
• C. $\displaystyle \dfrac{1}{2(\mathrm{x}-1)}+\dfrac{1}{2(\mathrm{x}+1)}-\dfrac{15}{\mathrm{x}+2}$
• D. $2+\dfrac{1}{2(x-1)}+\dfrac{1}{2(x+1)}-\dfrac{5}{x+2}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
If $\displaystyle \int\sqrt{x}(1-x^{3})^{-1/2}dx=\frac{2}{3}g(f(x))+c {\it}$ then
• A. $\displaystyle f(x)=\sqrt{x},g(x)=\sin^{-1}x$
• B. $\displaystyle f(x)=x^{2/3},g(x)=\cos^{-1}x$
• C. $\displaystyle f(x)=\sqrt{x},\ g(x)=\cos^{-1}x$
• D. $\displaystyle f(x)=x^{3/2},\ g(x)=\sin^{-1}x$

Given that for each $\displaystyle a \in (0, 1), \lim_{h \rightarrow 0^+} \int_h^{1-h} t^{-a} (1 -t)^{a-1}dt$ exists. Let this limit be $g(a)$
In addition, it is given that the function $g(a)$ is differentiable on $(0, 1)$