Mathematics

Single Correct Medium Published on 17th 09, 2020
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Realted Questions

Q1 Subjective Medium
Show that $$\displaystyle\int x^{3}\sqrt{q^{2}x^{8}-p^{2}}dx=\frac{1}{4q}\left [ \frac{t}{2}\sqrt{t^{2}-p^{2}}-\frac{1}{2}p^{2}\log \left ( t+\sqrt{t^{2}-p^{2}} \right ) \right ].$$ where $$t=qx^{4}$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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Q2 Single Correct Medium
If $$f(x) = \int^x_1 \dfrac{tan^{-1}t}{t} dt (x=0)$$, then the value of $$f(e^2)-f(\dfrac{1}{e^2})$$
  • A. $$2\pi$$
  • B. $$\dfrac{\pi}{2}$$
  • C. $$0$$
  • D. $$\pi$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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Q3 Single Correct Hard
$$\displaystyle\int{\frac{x^2(1-\ln{x})}{\ln^4{x}-x^4}dx}$$ is equal to
  • A. $$\displaystyle\frac{1}{2}\ln{\left(\frac{x}{\ln{x}}\right)}-\frac{1}{4}\ln{(\ln^2{x}-x^2)}+C$$
  • B. $$\displaystyle\frac{1}{4}\ln{\left(\frac{\ln(x)+x}{\ln{x}-x}\right)}-\frac{1}{2}\tan^{-1}{\left(\frac{\ln{x}}{x}\right)}+C$$
  • C. $$\displaystyle\frac{1}{4}\left(\ln{\left(\frac{\ln(x)-x}{\ln{x}+x}\right)}+\tan^{-1}{\left(\frac{\ln{x}}{x}\right)}\right)+C$$
  • D. $$\displaystyle\frac{1}{4}\ln{\left(\frac{\ln(x)-x}{\ln{x}+x}\right)}-\frac{1}{2}\tan^{-1}{\left(\frac{\ln{x}}{x}\right)}+C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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Q4 Subjective Medium
Evaluate :
$$\int \dfrac{sec^2x}{tanx}dx$$

Asked in: Mathematics - Integrals


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Q5 Passage Medium
Let $$\displaystyle I_{1}=\int_{0}^{1}(1-x^{2})^{1/3} dx$$  &  $$\displaystyle I_{2}=\int_{0}^{1}(1-x^{3})^{1/2} dx$$

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

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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