Solve ∫ (from 0 to log ( 1+ sqrt{ 2) of )}{left(e^x-e^x/2right)}^3{left(e^x-e^x/2right)}^11 wrt x

$$\int_{ 0 }^{ \log ( 1+ \sqrt{ 2 } ) } { \left( \frac{ { e }^{ x } - { e }^{ x } }{ 2 } \right) }^{ 3 } { \left( \frac{ { e }^{ x } - { e }^{ x } }{ 2 } \right) }^{ 11 } d x$$

Evaluate

$$0$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 11 to get 14.

$$\int _{0}^{\log_{10}\left(1+\sqrt{2}\right)}\left(\frac{e^{x}-e^{x}}{2}\right)^{14}\mathrm{d}x$$

Combine e^{x} and -e^{x} to get 0.

$$\int _{0}^{\log_{10}\left(1+\sqrt{2}\right)}\left(\frac{0}{2}\right)^{14}\mathrm{d}x$$

Zero divided by any non-zero number gives zero.

$$\int _{0}^{\log_{10}\left(1+\sqrt{2}\right)}0^{14}\mathrm{d}x$$

Calculate 0 to the power of 14 and get 0.

$$\int _{0}^{\log_{10}\left(1+\sqrt{2}\right)}0\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 0\mathrm{d}x$$

Find the integral of 0 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$0$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$0+0$$

Simplify.

$$0$$