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Wize University Calculus 1 Textbook > Integrals

Antiderivatives of Exponential Functions

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Antiderivatives (Indefinite Integral) of Exponential Functions

Let kk be a nonzero constant. Then we have the following indefinite integral rules:
∫ekxdx=ekxk+C\displaystyle\boxed{\int e^{kx}dx=\dfrac{e^{kx}}{k}+C}

∫akxdx=akxkln⁡a+C, (for a>0,a≠1)\displaystyle\boxed{\displaystyle \int a^{kx}dx=\dfrac{a^{kx}}{k\ln a}+C},\ \text{(for}\ a>0,a\neq1)

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Example: Exponential Indefinite Integrals

Evaluate the following Indefinite Integral

∫(ex)9dx\displaystyle \int (e^x)^9dx

∫(ex)9dx\displaystyle \int (e^x)^9dx

=∫e9xdx=\displaystyle \int e^{9x}dx

=e9x9+C\displaystyle =\frac{e^{9x}}{9}+C
Evaluate the following indefinite integral

∫(e3−e3x+3x)dx\displaystyle \int( e^3-e^{3x}+3^x )dx

Evaluate the following indefinite integral

∫7(x2+1)dx\displaystyle \int 7^{\left(\displaystyle \frac{x}{2}+1\right)}dx
Extra Practice
Evaluate the indefinite integral ∫6x+e−6x−3x2+1dx\displaystyle \int6^x+e^{-6x}-\frac{3}{x^2+1}dx.
Evaluate the indefinite integral ∫6x+e−6x−3x2+1dx\displaystyle \int6^x+e^{-6x}-\frac{3}{x^2+1}dx.
Practice: Definite Integral (~F2017 Final Q26)
Practice: Definite Integral
Evaluate ∫0319+x2+2xln⁡2 dx\int_0^3\frac{1}{9+x^2}+2^x\ln2\ dx.
Antiderivatives: Exponential Functions
Find the indefinite integral ∫(6x+e−6x) dx\displaystyle \int(6^x+e^{-6x}) \ dx
Exponential Functions: $f$ from $f''$
Given f′′(x)=5ex+2sin⁡xf''(x)=5e^x+2\sin x and given f(0)=0f(0)=0, f(π)=0f(\pi)=0, find f(x)f(x). Enter "f(x)=..."
Indefinite Integrals with Exponential
Compute the following integral: ∫3−x dx\displaystyle \int 3^{-x}\,\text{d}x.
Use upper case CC to denote any constants.
Antiderivatives: Exponential Functions
Which of the following is the most general antiderivative of the function e3x+7e^{3x+7}? In the functions below, cc is an arbitrary constant.
Find ∫(cos7)e3xdx\displaystyle \int (cos7)e^{3x}dx
Antiderivatives: Exponential Functions
Find an antiderivative of f(x)=2e3x−2f(x) = 2e^{3x-2}
Practice: Definite Integral
Evaluate ∫0319+x2+2xln⁡2 dx\displaystyle \int_0^3\frac{1}{9+x^2}+2^x\ln2\ dx.
Find the equation of the tangent line of the graph y=excos⁡ xy=\frac{e^x}{\cos\ x}at the point where 𝑥 = 0.
Exponential Functions: $f$ from $f''$
Given f′′(x)=5ex+2sin⁡xf''(x)=5e^x+2\sin x and given f(0)=0f(0)=0, f(π)=0f(\pi)=0, find f(x)f(x). Enter "f(x)=..."
Indefinite Integrals with Exponential
Compute the following integral: ∫3−x dx\displaystyle \int 3^{-x}\,\text{d}x.
Use upper case CC to denote any constants.
Antiderivatives: Exponential Functions
Find the indefinite integral ∫(6x+e−6x) dx\displaystyle \int(6^x+e^{-6x}) \ dx
Find the equation of the tangent line to the curve y=23e3−3xy=2^3e^{3-3x} at x=1x=1 .
Enter your answer as y=...
Evaluate the indefinite integral ∫6x+e−6x−3x2+1dx\displaystyle \int6^x+e^{-6x}-\frac{3}{x^2+1}dx.
Higher Order Derivatives
Find a general formula for the nth-derivative of f(x)=e3xf(x)=e^{3x}.
Antiderivatives: Trigonometric and Exponential Functions
If f′′(x)=cos⁡(2x)−x4+exf''\left(x\right)=\cos\left(2x\right)-x^4+e^x, f(0)=0=f′(0)f\left(0\right)=0=f'\left(0\right)
Practice: Definite Integral
Practice: Definite Integral
Evaluate ∫0319+x2+2xln⁡2 dx\int_0^3\frac{1}{9+x^2}+2^x\ln2\ dx.