# Formulas related to Integrals of Exponential Functions

$$\int e^{cx}dx = \frac{1}{c}e^{cx}$$ $$\int a^{cx}dx = \frac{1}{c\cdot \ln a}a^{cx}, (\text{for } a>0, a\ne1 )$$ $$\int x \cdot e^{cx} = \frac{e^{cx}}{c^2}(cx-1)$$ $$\int x^2 \cdot e^{cx} = e^{cx}\left(\frac{x^2}{c}-\frac{2x}{c^2} + \frac{2}{c^3}\right)$$ $$\int x^n \cdot e^{cx}dx = \frac{1}{c}x^ne^{cx}-\frac{n}{c}\int x^{n-1}e^{cx} dx$$ $$\int \frac{e^{cx}}{x} dx = \ln|x| + \sum\limits_{i=1}^\infty \frac{(cx)^i}{i \cdot i!}$$ $$\int \frac{e^{cx}}{x^n} = \frac{1}{n-1}\left(-\frac{e^{cx}}{x^{n-1}} + c\cdot \int \frac{e^{cx}}{x^{n-1}} dx \right)$$ $$\int e^{cx}\cdot \ln x dx = \frac{1}{c} e^{cx}\ln|x| + E_{\,i}(cx)$$ $$\int e^{cx}\cdot \sin(bx) dx = \frac{e^{cx}}{c^2 + b^2} \left(c\cdot \sin(bx) - b\cdot cos(bx)\right)$$ $$\int e^{cx}\cdot \cos(bx) dx = \frac{e^{cx}}{c^2 + b^2} \left(c\cdot \sin(bx) + b\cdot \cos(bx)\right)$$ $$\int e^{cx}\cdot \sin^nx dx = \frac{e^{cx}\cdot \sin^{n-1}x}{c^2 + n^2} (c\cdot \sin x - n\cdot \cos(bx)) + \frac{n(n-1)}{c^2 + n^2} \int e^{cx} \sin^{n-2} dx$$ $$\int e^{cx}\cdot \cos^nx dx = \frac{e^{cx}\cdot \cos^{n-1}x}{c^2 + n^2} (c\cdot \sin x + n\cdot \cos(bx)) + \frac{n(n-1)}{c^2 + n^2} \int e^{cx} \cos^{n-2} dx$$