1. Tardigrade
  2. Question
  3. Mathematics
  4. The value of ∫( sin 2x cos 2x/( sin 3x+ cos 3x)2)dx is

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. The value of $ \int_{{}}^{{}}{\frac{{{\sin }^{2}}x{{\cos }^{2}}x}{{{({{\sin }^{3}}x+{{\cos }^{3}}x)}^{2}}}dx} $ is

Bihar CECEBihar CECE 2015

Solution:

Let $ I=\int_{{}}^{{}}{\frac{{{\sin }^{2}}x{{\cos }^{2}}x}{{{({{\sin }^{3}}x+{{\cos }^{3}}x)}^{2}}}}dx $ $ \int_{{}}^{{}}{\frac{{{\tan }^{2}}x{{\sec }^{2}}x}{{{(1+{{\tan }^{3}}x)}^{2}}}dx} $ [dividing numerator and denominator by $ {{\cos }^{6}}x $ ] $ =\frac{1}{3}\int_{{}}^{{}}{\frac{3{{\tan }^{2}}x{{\sec }^{2}}x}{{{(1+{{\tan }^{3}}x)}^{2}}}}dx $ $ =\frac{1}{3}\int_{{}}^{{}}{\frac{1}{{{(1+{{\tan }^{3}}x)}^{2}}}}d(1+{{\tan }^{3}}x) $ $ =-\frac{1}{3(1+{{\tan }^{2}}x)}+C $