Mathematics

$$\int \frac{1}{x(x^{n}-1)}dx$$


ANSWER

$$\frac{1}{n} log\left | \frac{x^{n}}{x^{n}+1} \right |+ c$$


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

Q1 Single Correct Medium
Evaluate $$\displaystyle \int \frac{1}{(x-1)(x^{2}+1)}dx$$
  • A. $$\displaystyle \frac{1}{2}log(x-1)+\frac{1}{4}log(x^{2}+1)-\frac{1}{2}\tan ^{-1}x+c$$
  • B. $$\displaystyle \frac{1}{2}log(x-1)-\frac{1}{2}log(x^{2}+1)-\frac{1}{2}\tan ^{-1}x+c$$
  • C. $$\displaystyle \frac{1}{2}log(x-1)-\frac{1}{4}log(x^{2}+1)+\tan ^{-1}x+c$$
  • D. $$\displaystyle \frac{1}{2}log(x-1)-\frac{1}{4}log(x^{2}+1)-\frac{1}{2}\tan ^{-1}x+c$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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Q2 Single Correct Medium
$$\int _{-1  }^{ 1 }\dfrac {{\sqrt {1+x+x^2}- \sqrt { 1-x+x^2 }  }  }{\sqrt { 1+x+x^2  }+\sqrt {  1-x+x^2} } dx=$$
  • A. $$\dfrac {3\pi}{2}$$
  • B. $$\dfrac {\pi}{2}$$
  • C. $$-1$$
  • D. $$0$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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Q3 Single Correct Medium
$$\displaystyle \int\frac{x}{(x^{2}-a^{2})(x^{2}-b^{2})}dx=$$
  • A. $$\displaystyle \frac{1}{2(a^{2}-b^{2})}\log|\frac{x^{2}-b^{2}}{x^{2}-a^{2}}|+c$$
  • B. $$\displaystyle \log|\frac{x^{2}-a^{2}}{x^{2}-b^{2}}|+c$$
  • C. $$\displaystyle \frac{1}{(a^{2}-b^{2})}\log|\frac{x^{2}-a^{2}}{x^{2}-b^{2}}|+c$$
  • D. $$\displaystyle \frac{1}{2(a^{2}-b^{2})}\log|\frac{x^{2}-a^{2}}{x^{2}-b^{2}}|+c$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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Q4 Subjective Medium
Integrate the function    $$\displaystyle \frac {3x^2}{x^6+1}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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Q5 Passage Hard
Let us consider the integral of the following forms
$$f{(x_1,\sqrt{mx^2+nx+p})}^{\tfrac{1}{2}}$$
Case I If $$m>0$$, then put $$\sqrt{mx^2+nx+C}=u\pm x\sqrt{m}$$
Case II If $$p>0$$, then put $$\sqrt{mx^2+nx+C}=u\pm \sqrt{p}$$
Case III If quadratic equation $$mx^2+nx+p=0$$ has real roots $$\alpha$$ and $$\beta$$, then put $$\sqrt{mx^2+nx+p}=(x-\alpha)u\:or\:(x-\beta)u$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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