Mathematics

Evaluate the following integrals:
$$\displaystyle \int { \cfrac { 1 }{ { x }^{ 2/3 }\sqrt { { x }^{ 2/3 }-4 }  }  } dx\quad $$


SOLUTION
$$I=\displaystyle\int{\dfrac{dx}{{x}^{\frac{2}{3}}\sqrt{{x}^{\frac{2}{3}}-4}}}$$

$$=\displaystyle\int{\dfrac{{x}^{-\frac{2}{3}}dx}{\sqrt{{x}^{\frac{2}{3}}-4}}}$$
Let $$t={x}^{\frac{1}{3}}\Rightarrow\,dt=\dfrac{1}{3}{x}^{\frac{1}{3}-1}dx$$

$$\Rightarrow\,dt=\dfrac{1}{3}{x}^{\frac{-2}{3}}dx$$

$$\Rightarrow\,3\,dt={x}^{\frac{-2}{3}}dx$$

$$\therefore\,I=3\displaystyle\int{\dfrac{dt}{\sqrt{{t}^{2}-{2}^{2}}}}$$

We know that $$\displaystyle\int{\dfrac{dx}{\sqrt{{x}^{2}-{a}^{2}}}}=\log{\left|x+\sqrt{{x}^{2}-{a}^{2}}\right|}+c$$
Replace $$x\rightarrow\,t$$ and $$a\rightarrow\,2$$

$$=3\log{\left|t+\sqrt{{t}^{2}-{2}^{2}}\right|}+c$$

$$=3\log{\left|t+\sqrt{{t}^{2}-4}\right|}+c$$

$$=3\log{\left|{x}^{\frac{1}{3}}+\sqrt{{\left({x}^{\frac{1}{3}}\right)}^{2}-4}\right|}+c$$ where $$t={x}^{\frac{1}{3}}$$

$$=3\log{\left|{x}^{\frac{1}{3}}+\sqrt{{{x}^{\frac{2}{3}}-4}}\right|}+c$$

View Full Answer

Its FREE, you're just one step away


Subjective Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 111
Enroll Now For FREE

Realted Questions

Q1 Subjective Medium
Solve:
$$\int\limits_0^{\frac{\pi }{2}} {\frac{{x\sin 2xdx}}{{{{\cos }^4}x + {{\sin }^4}x}}} $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Hard
$$\int_{}^{} {\frac{{dx}}{{x\left( {{x^n} + 1} \right)}}} $$ is equal to
  • A. $$\frac{1}{n}\log \left( {\frac{{{x^n}}}{{{x^n} + 1}}} \right) + c$$
  • B. $$\log \left( {\frac{{{x^n}}}{{{x^n} + 1}}} \right) + c$$
  • C. None of these
  • D. $$-\frac{1}{n}\log \left( {\frac{{{x^n} + 1}}{{{x^n}}}} \right) + c$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
Evaluate:
$$\int _{ 0 }^{ \pi /4 }{ \left( { sin }^{ 2 }\dfrac { x }{ 2 } -cos^{ 2 }\dfrac { x }{ 2 }  \right)  } dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
Solve:$$\int(1-\cos x)cosec^2 x dx=$$ ?

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Medium
Evaluate : $$\displaystyle \int e^{x} \dfrac {\log (\sin e^{x})}{\tan e^{x}} dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer