Subject: Calculus

# List Of Integrals Of Exponential Functions

 The Page at a Glance

## Integrals involving ordinary exponential functions

• \int e^{x}dx = e^{x} + C
• \int e^{bx}dx = \frac{1}{b}e^{bx} + C
where b is a constant
• \int a^{x}dx = \frac{1}{\ln{a}}\cdot a^{x} + C
where a is a constant
• \int a^{bx}dx = \frac{1}{b\cdot \ln{a}}\cdot a^{bx} + C
where a and b are constants
• \int xe^{x}dx = e^{x}\left ( x - 1 \right ) + C
• \int xe^{bx}dx = \frac{1}{b^2}\cdot e^{bx}\left ( cx - 1 \right ) + C
where b is a constant
• \int x^2e^{bx}dx = \frac{1}{b^{2}}\cdot e^{bx}\left ( bx^2 -2x + \frac{2}{b} \right ) + C
where b is a constant
• \int x^{n}e^{x}dx = x^n e^x - n\int x^{n-1}e^{x}dx
• \int x^{n}e^{bx}dx = \frac{1}{b}x^n e^{bx} - \frac{n}{b}\int x^{n-1}e^{bx}dx
where b is a constant
• \int \frac{e^{bx}}{x^n}dx = \frac{1}{n-1}\left ( -\frac{e^{bx}}{x^{n-1}} + b\int \frac{e^{bx}}{x^{n-1}}dx \right )
where b is a constant

## Integrals of exponential functions containing trigonometric forms

• \int e^{x}\sin{x}dx = \frac{1}{2}\cdot e^{x}\left (\sin{x} - \cos{x} \right )+ C
• \int e^{ax}\sin{bx}dx = \frac{1}{a^2+b^2}\cdot e^{ax}\left (a\sin{bx} - b\cos{bx} \right ) + C
where a and b are constants
• \int e^{x}\cos{x}dx = \frac{1}{2}\cdot e^{x}\left (\sin{x} + \cos{x} \right )+ C
• \int e^{ax}\cos{bx}dx = \frac{1}{a^2+b^2}\cdot e^{ax}\left (b\sin{bx} + a\cos{bx} \right ) + C
where b is a constant
• \int e^{ax}\sin^{r}{bx}dx = \frac{e^{ax}\sin^{r-1}{x}}{a^2+r^2}\left (a\sin{x} - r\cos{x} \right ) + \frac{r(r-1)}{a^2+r^2} \int e^{ax}\sin^{r-2}{x}dx
where a is a constant and r is an integer
• \int e^{ax}\cos^{r}{bx}dx = \frac{e^{ax}\cos^{r-1}{x}}{a^2+r^2}\left (a\cos{x} + r\sin{x} \right ) + \frac{r(r-1)}{a^2+r^2} \int e^{ax}\cos^{r-2}{x}dx
where a is a constant and r is an integer