Wize University Calculus 1 Textbook > Integrals

Antiderivatives of Exponential Functions

Antiderivatives (Indefinite Integral) of Exponential Functions

Let $k$ be a nonzero constant. Then we have the following indefinite integral rules:

\displaystyle\boxed{\int e^{kx}dx=\dfrac{e^{kx}}{k}+C}

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

Example: Exponential Indefinite Integrals

Evaluate the following Indefinite Integral

$\displaystyle \int (e^x)^9dx$

$=\displaystyle \int e^{9x}dx$

$\displaystyle =\frac{e^{9x}}{9}+C$

Evaluate the following indefinite integral

$\displaystyle \int( e^3-e^{3x}+3^x )dx$

Don't forget $+C$!

I don't know

Evaluate the following indefinite integral

$\displaystyle \int 7^{\left(\displaystyle \frac{x}{2}+1\right)}dx$

$\displaystyle \frac{7^{\frac{1}{2}x}\times7}{\frac{1}{2}\times\ln 7}$

$\displaystyle \displaystyle \frac{14\times7^{\frac{1}{2}x}}{\ln 7}+C$

$\displaystyle \displaystyle \frac{7^{\frac{1}{2}x}}{\ln 7}+C$

$\displaystyle \displaystyle \frac{7\times7^{\frac{1}{2}x}}{x\times\ln 7}+C$

I don't know

Extra Practice

Practice: Definite Integral

Evaluate $\int_0^3\frac{1}{9+x^2}+2^x\ln2\ dx$ .

Antiderivatives: Trigonometric and Exponential Functions

If $f''\left(x\right)=\cos\left(2x\right)-x^4+e^x$ , $f\left(0\right)=0=f'\left(0\right)$

Higher Order Derivatives

Find a general formula for the nth-derivative of $f(x)=e^{3x}$ .

Evaluate the indefinite integral $\displaystyle \int6^x+e^{-6x}-\frac{3}{x^2+1}dx$ .

Find the equation of the tangent line to the curve $y=2^3e^{3-3x}$ at $x=1$ .

Enter your answer as y=...

Antiderivatives: Exponential Functions

Find the indefinite integral $\displaystyle \int(6^x+e^{-6x}) \ dx$

Indefinite Integrals with Exponential

Compute the following integral: $\displaystyle \int 3^{-x}\,\text{d}x$ .

Use upper case $C$ to denote any constants.

Exponential Functions: $f$ from $f''$

Given $f''(x)=5e^x+2\sin x$ and given $f(0)=0$ , $f(\pi)=0$ , find $f(x)$ . Enter "f(x)=..."

Find the equation of the tangent line of the graph $y=\frac{e^x}{\cos\ x}$ at the point where 𝑥 = 0.

Practice: Definite Integral

Evaluate $\displaystyle \int_0^3\frac{1}{9+x^2}+2^x\ln2\ dx$ .

Antiderivatives: Exponential Functions

Find an antiderivative of $f(x) = 2e^{3x-2}$

Find $\displaystyle \int (cos7)e^{3x}dx$

Antiderivatives: Exponential Functions

Which of the following is the most general antiderivative of the function $e^{3x+7}$ ? In the functions below, $c$ is an arbitrary constant.

Indefinite Integrals with Exponential

Compute the following integral: $\displaystyle \int 3^{-x}\,\text{d}x$ .

Use upper case $C$ to denote any constants.

Exponential Functions: $f$ from $f''$

Given $f''(x)=5e^x+2\sin x$ and given $f(0)=0$ , $f(\pi)=0$ , find $f(x)$ . Enter "f(x)=..."

Antiderivatives: Exponential Functions

Find the indefinite integral $\displaystyle \int(6^x+e^{-6x}) \ dx$

Practice: Definite Integral (~F2017 Final Q26)

Practice: Definite Integral

Evaluate $\int_0^3\frac{1}{9+x^2}+2^x\ln2\ dx$ .

Evaluate the indefinite integral $\displaystyle \int6^x+e^{-6x}-\frac{3}{x^2+1}dx$ .

Antiderivatives of Exponential Functions

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Antiderivatives (Indefinite Integral) of Exponential Functions

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Example: Exponential Indefinite Integrals

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Extra Practice

Practice: Definite Integral

Practice: Definite Integral

Practice: Definite Integral