1. Tardigrade
  2. Question
  3. Mathematics
  4. The value of the definite integral ∫ limits-11 e - x 4(1+ ln ( x +√ x 2+1)+5 x 3-4 x 4) dx is equal to

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. The value of the definite integral $\int\limits_{-1}^1 e ^{- x ^4}\left(1+\ln \left( x +\sqrt{ x ^2+1}\right)+5 x ^3-4 x ^4\right) dx$ is equal to

Integrals

Solution:

As, $f(x)=1+\ln \left(x+\sqrt{x^2+1}\right)+5 x^3-4 x^4$
Clearly, $\ln \left(\sqrt{x^2+1}+x\right), 5 x^3$ are odd functions, so
$I=\int\limits_{-1}^1\left(1-4 x^4\right) e^{-x^4} d x=2 \int\limits_0^1\left(1-4 x^4\right) e^{-x^4} d x=2 \int\limits_0^1 \frac{d}{d x}\left(x \cdot e^{-x^4}\right) d x=2\left(\frac{x}{e^{x^4}}\right)_0^1=2\left(\frac{1}{e}-0\right)=\frac{2}{e}$