Solve int ∫ (from - pi to pi) of ( e^2x)sin(nx) wrt x

$$\int \int _ { - \pi } ^ { \pi } ( e ^ { 2 x } ) \sin ( n x ) d x$$

Evaluate

$$\frac{x\left(n\cos(\pi n)-\left(e^{\pi }\right)^{4}n\cos(\pi n)+2\left(e^{\pi }\right)^{4}\sin(\pi n)+2\sin(\pi n)\right)}{\left(e^{\pi }\right)^{2}\left(n^{2}+4\right)}+С$$

Solution Steps

Find the integral of \int _{-\pi }^{\pi }e^{2x}\sin(nx)\mathrm{d}x using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\int _{-\pi }^{\pi }e^{2x}\sin(nx)\mathrm{d}xx$$

Simplify.

$$\frac{\left(2\sin(n\pi )\left(e^{4\pi }+1\right)+n\cos(n\pi )\left(-e^{4\pi }+1\right)\right)x}{\left(4+n^{2}\right)e^{2\pi }}$$

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.

$$\frac{\left(2\sin(n\pi )\left(e^{4\pi }+1\right)+n\cos(n\pi )\left(-e^{4\pi }+1\right)\right)x}{\left(4+n^{2}\right)e^{2\pi }}+С$$

Differentiate w.r.t. x

$$\frac{n\cos(\pi n)-e^{4\pi }n\cos(\pi n)+2e^{4\pi }\sin(\pi n)+2\sin(\pi n)}{e^{2\pi }\left(n^{2}+4\right)}$$