1. Tardigrade
  2. Question
  3. Mathematics
  4. ∫-π π ( sin 4x/ sin 4x+ cos 4x)dx is equal to:

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. $ \int_{-\pi }^{\pi }{\frac{{{\sin }^{4}}x}{{{\sin }^{4}}x+{{\cos }^{4}}x}}dx $ is equal to:

KEAMKEAM 2005

Solution:

Let $ I=\int_{-\pi }^{\pi }{\frac{{{\sin }^{4}}xdx}{{{\sin }^{4}}x+{{\cos }^{4}}x}} $
$ =4\int_{0}^{\pi }{\frac{{{\sin }^{4}}x}{{{\sin }^{4}}x+{{\cos }^{4}}x}}dx $
$ I=4\int_{0}^{\pi /2}{\frac{{{\sin }^{4}}x}{{{\sin }^{4}}x+{{\cos }^{4}}x}}dx $ ..(i)
$ I=4\int_{0}^{\pi /2}{\frac{{{\cos }^{4}}x}{{{\sin }^{4}}x+{{\cos }^{4}}x}}dx $ ..(ii)
On adding Eqs. (i) and (ii), we get
$ 2I=4\int_{0}^{\pi /2}{1.}\,dx=2\pi $