Question

The value of \( \int_{-1 / 8}^{1 / 8} \cos ^{-1}\left(x^{5}+5 x\right) d x \) is (a) \( \frac{\pi}{2} \) (b) \( \sqrt{2 \pi} \) (c) \( \frac{\pi}{8} \) (d) None of these

Answer 1

Answer: Option C: \(π\over8\)

Solution:

Use,  \(\int\limits_a^b\)  f(x) .dx = \(\int\limits_a^b\) f(b+a-x) .dx, 

⇒ here a = -\(1\over8\) and b = \(1\over8\),

⇒ I = \(\int\) cos^-1(x⁵+5x) .dx 

= \(\int\) cos^-1[\(1\over8\)+(-\(1\over8\)) - (x⁵+5x)] .dx  

= \(\int\) cos^-1(-(x⁵+5x)) .dx 

I = \(\int\) π - cos^-1(x⁵+5x) .dx ⇔ {cos^-1(-x) = π - cos^-1(x)}

⇒ 2I = I + I =   \(\int\) cos^-1(x⁵+5x) .dx +   \(\int\) π - cos^-1(x⁵+5x) .dx

=   \(\int\) cos^-1(x⁵+5x) + π - cos^-1(x⁵+5x) .dx

2I = \(\int\) π .dx 

= [πx] + c (from  -\(1\over8\) to \(1\over8\))

2I = \(π\over8\)- (-\(π\over8\)) = \(2π\over8\) 

⇒ I = \(2π\over8\).\(1\over2\) 

⇒ \(π\over8\) 

Option C is the correct answer.