The value of ∫_-1^1(x^6)/(6^x+1) dx is

The correct answer is Option (4) → \( \displaystyle\frac{1}{7} \)

Let \( I=\int_{-1}^{1}\frac{x^{6}}{6^{x}+1}\,dx \). Since the limits are symmetric, replacing x by −x gives

    \( \displaystyle I=\int_{-1}^{1}\frac{x^{6}}{6^{-x}+1}\,dx \)

    \( \displaystyle =\int_{-1}^{1}\frac{6^{x}x^{6}}{1+6^{x}}\,dx \)

Adding the two forms of I,

    \( \displaystyle 2I=\int_{-1}^{1}\frac{x^{6}(1+6^{x})}{6^{x}+1}\,dx \)

    \( \displaystyle =\int_{-1}^{1}x^{6}\,dx \)

    \( \displaystyle =2\int_{0}^{1}x^{6}\,dx \)

    \( \displaystyle =\frac{2}{7} \)

Hence \( I=\frac{1}{7} \).