Solve ∫ log(m^2+4m-5)^pi/sin(logpi/2)^cospi

$$\int \log \frac { ( m ^ { 2 } + 4 m - 5 ) ^ { \pi } } { \sin ( \log \pi / 2 ) ^ { \cos \pi } }$$

Evaluate

$$\log(e)x\left(\pi \ln(\left(m-1\right)\left(m+5\right))+\ln(0.19486506597972128)\right)+С$$

Solution Steps

Evaluate trigonometric functions in the problem

$$\int {\log_{10} {(\frac{{(m ^ {2} + 4 m - 5)} ^ {\pi}}{0.19486506597972128 ^ {\cos(\pi)}})}} dx$$

Find the integral of \log_{10}\left(\frac{\left(m^{2}+4m-5\right)^{\pi }}{0.19486506597972128^{\cos(\pi )}}\right) using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\log_{10}\left(\frac{\left(m^{2}+4m-5\right)^{\pi }}{0.19486506597972128^{\cos(\pi )}}\right)x$$

Simplify.

$$\frac{\left(\pi \ln(m^{2}+4m-5)+\ln(\frac{608953331186629}{3125000000000000})\right)x}{\ln(10)}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\frac{\left(\pi \ln(m^{2}+4m-5)+\ln(\frac{608953331186629}{3125000000000000})\right)x}{\ln(10)}+С$$

Differentiate w.r.t. x

$$\log(e)\left(\pi \ln(\left(m-1\right)\left(m+5\right))+\ln(0.19486506597972128)\right)$$