Mathematics

$$\int { { e }^{ 3x } } \cdot { x }^{ 2 }dx$$


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Q1 Subjective Medium
Integrate the function    $$\tan^{-1}x$$

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Q2 Single Correct Hard
The value of $$\int {{{(x - 2)} \over {{{\left\{ {{{(x - 2)}^2}{{(x + 3)}^7}} \right\}}^{{1 \over 3}}}}}} dx$$ is 

  • A. $${3 \over {20}}{\left( {{{x - 2} \over {x + 3}}} \right)^{{3 \over 4}}} + c$$
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Q3 Single Correct Medium
If $$\displaystyle \int_0^{\pi} \dfrac{x^2}{(1+\sin \,x)^2} dx = A$$ the $$\displaystyle \int_0^{\pi} \dfrac{2x^2cos^2(x/2)}{(1+\sin x)^2} dx=$$ ?
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Q4 Single Correct Medium
$$\displaystyle \int \sinh^{-1}\left(\frac{x}{4}\right)dx$$ is equal to
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  • C. $$x\sinh^{-1} {\left(\dfrac{x}{2}\right)-x}\sqrt{x^{2}+16}+c$$
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Q5 Passage Medium
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 $$\displaystyle f(x)>0    \forall  x \in  R$$

On the basis of above information, answer the following questions :

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