If f(x)=∫ 1/x^(1/4)(1+x^(1/4))dx,f(0)=-6, then f(1) is equal to :

Solution

Maths sir · 5 steps

Put x = t⁴ so every fourth root becomes a plain power of t, then divide the fraction out.

1. Substitute. Let x = t⁴, so x^(1/4) = t and dx = 4t³ dt. The integrand becomes 4t³/t(1+t) = 4t²/(1+t).
2. Divide out. The top has a higher degree than the bottom, so divide first. Writing t² = (t² - 1) + 1 gives the split below.

   4t²/(1+t) = 4(t - 1 + 1/(1+t))

3. Integrate. f = 4(t²/2 - t + ln(1+t)) + C = 2t² - 4t + 4 ln(1+t) + C, with t = x^(1/4).
4. Fix the constant. At x = 0, t = 0 and every term is zero, so f(0) = C = -6.
5. So. At x = 1, t = 1: f(1) = 2 - 4 + 4 ln 2 - 6 = 4 ln 2 - 8 = 4(logₑ 2 - 2), which is option (a).

The idea to keep. When an integrand mixes x^(1/n) terms, put x = tⁿ to clear the roots. If the result is a fraction with a higher power on top, divide before you integrate.

Answer from the NTA final answer key. Solution by PrepWiser.