Mathematics

# Integrate :$\dfrac{1}{1-\cot x}$

$\dfrac {1}{2}\log |\sin x-\cos x|+\dfrac {1}{2}x+C$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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Q4 Single Correct Medium
If $\displaystyle f\left ( x \right )$ is a function of $x$ such that $\displaystyle \frac{1}{\left ( 1 + x \right ) \left ( 1 + x^{2} \right )} = \frac{A}{1 + x} + \frac{f\left ( x \right )}{1 + x^{2}}$ for all $\displaystyle x \: \epsilon \: R$ then $\displaystyle f\left ( x \right )$ is
• A. $\displaystyle \frac{x + 1}{2}$
• B. $\displaystyle 1 - x$
• C. none of these
• D. $\displaystyle \frac{1 - x}{2}$

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$