Mathematics

Evaluate: $$\displaystyle \int_{0}^{\frac{\pi}{2}} \dfrac{\sin x \cdot \cos x}{1+\sin^4 x}\cdot dx$$.


SOLUTION
$$\displaystyle \overset{\pi/2}{\underset{0}{\int}} \frac{\sin x \cdot \cos x}{1 + \sin^4 x} \cdot dx$$.
Let $$\sin^2 n = t$$
$$2 \sin x \,\cos x \,dx = dt$$
When $$x = 0, t = 0$$

           $$x = \dfrac{\pi}{2}, t = 1$$

$$= \dfrac{1}{2} \displaystyle \overset{dt/2}{\underset{1 + t^2}{\int}} \frac{dt}{1 + t^2}$$

$$= \dfrac{1}{2} \displaystyle \overset{1}{\underset{0}{\int}} \frac{dt}{1 + t^2}$$

$$= \dfrac{1}{2} [\tan^{-1} - \tan^{-1} 0]$$

$$= \dfrac{1}{2} \left[\dfrac{\pi}{4} - 0 \right]$$

$$= \dfrac{\pi}{8}$$
View Full Answer

Its FREE, you're just one step away


Subjective Hard Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 114
Enroll Now For FREE

Realted Questions

Q1 Single Correct Hard
$$\displaystyle \int e^x\left(\frac{1+\sin\,x}{1+\cos\,x}\right)dx$$ is equal to
  • A. $$e^x\sec^2\dfrac{x}{2}+C$$
  • B. $$e^x\sec\dfrac{x}{2}+C$$
  • C. $$e^x+\tan\,x+C$$
  • D. $$e^x\tan\dfrac{x}{2}+C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Hard
$$\displaystyle \int_{0}^{4}\frac{(y^2-4y+5)\sin (y-2)dy}{[2y^2-8y+11]}$$ is equal to
  • A. 2
  • B. -2
  • C. none of these
  • D.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
Find $$\displaystyle \int \dfrac{x^{3}+x+1}{x^{2}-1}\ dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
Solve : 
$$\displaystyle \int \dfrac{u}{v} dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Medium
$$\int \frac{x}{x^2 + a^2} \;dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer