Mathematics

$$\displaystyle\int \sec^{\dfrac{2}{3}}x cosec^{\dfrac{4}{3}}xdx$$ is equal to?


ANSWER

$$-3\cot^{\dfrac{1}{3}}x+c$$


SOLUTION
$$I=\displaystyle\int (\sec x)^{2/3}\cdot(cosec x)^{4/3}dx$$
$$=\displaystyle\int\dfrac{1}{(\sin x)^{4/3}\cdot (\cos x)^{2/3}}dx$$
Multiplying numerator and denominator by $$cosec^2x$$, we get
$$I=\displaystyle\int \dfrac{cosec^2x}{(cot x)^{2/3}}dx$$
Let $$\cot x=t^3$$
$$\Rightarrow cosec^2xdx=-3t^2dt$$
Hence $$I=-3\displaystyle\int \dfrac{t^2dt}{t^2}=-3t+C=-3(\cot x)^{1/3}+C$$.
View Full Answer

Its FREE, you're just one step away


Single Correct Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86
Enroll Now For FREE

Realted Questions

Q1 Subjective Medium
Evaluate $$\displaystyle\int^4_1\sqrt{x}dx$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Medium
Integrate the following function with respect to $$x$$
$$x^{2}.\cos{(x^{3})}\sqrt{\sin^{7}{(x^{3})}}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 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
Q4 Subjective Medium
Evaluate the given expression $$\displaystyle\int\dfrac{dx}{9+16\sin^2x}$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Single Correct Medium
$$\int\limits_0^\pi  {\log (1 + cosx)dx = } $$
  • A. $$ - \frac{\pi }{2}\log 2$$
  • B. $$ - \frac{\pi }{3}$$
  • C. $$ - 2\pi \log 2$$
  • D. $$ - \pi \log 2$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer