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Wize University Calculus 1 Textbook > Integrals

Antiderivatives & The Indefinite Integrals

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Antiderivatives (Indefinite Integral)

If F′(x)=f(x)F^{\prime}(x)=f(x), then F(x)F(x) is called the antiderivative or indefinite integral of f(x)f(x).

Wize Tip
Antiderivative just means reversing a derivative.

Indefinite Integrals

There are many possible antiderivatives to one function f(x)f(x) . The collection of all such antiderivatives is called the indefinite integral of f(x)f(x) , and is denoted
∫f(x)dx\int f(x)dx
If F(x)F(x) is an antiderivative of f(x)f(x), then
∫f(x)dx=F(x)+C\int_{ }^{ }f(x)dx=F(x)+C
where CC is an arbitrary constant.


Watch Out!
When asked to find the antiderivative of a function, don’t forget the +C+C!


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Reverse Power Rule

If f(x)=xnf(x)=x^n , when n≠−1n\ne-1, then we can use the power rule in revere to find its indefinite integral:
∫xndx=xn+1n+1+C\boxed{\int x^ndx=\frac{x^{n+1}}{n+1}+C}

Watch Out!
1x=x−1\displaystyle \frac{1}{x}=x^{-1} does not follow the Reverse Power Rule!

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Example: Antiderivatives

Find all the antiderivatives of f(x)=4x3+xf(x)=4x^3+\sqrt{x}

f(x)=4x3+x1/2⇓F(x)=4x44+x3/213/2+CF(x)=x4+23x3/2+C\begin{array}{c} f(x) = 4x^3+x^{1/2}\\ \Downarrow\\ F(x) = \dfrac{4x^4}{4}+x^{3/2}\dfrac{1}{3/2}+C\\ \\ {{F(x) = x^4+\dfrac{2}{3}x^{3/2}+C}} \end{array}

Find the antiderivative of f(x)f(x).
f(x)=4x+x22+5\displaystyle f(x)=4x+\frac{x^2}{2}+5
Evaluate the following indefinite integral

∫(x+2)2dx\displaystyle \int(x+2)^2dx
Evaluate the following indefinite integral

∫x4−3xxdx\displaystyle \int \frac{x^4-3x}{x}dx
Extra Practice
Practice: Definite Integral
Practice: Definite Integral
Evaluate ∫−20∣x2−1∣dx\displaystyle \int_{-2}^0\left|x^2-1\right|dx.
Antiderivatives: Indefinite integrals
Evaluate the integral ∫1+1x+2x2+3x3dx\displaystyle\int_{ }^{ }1+\frac{1}{x}+\frac{2}{x^2}+\frac{3}{x^3}dx.
Antiderivatives: Indefinite integrals
Evaluate the integral ∫1+1x+2x2+3x3dx\displaystyle\int_{ }^{ }1+\frac{1}{x}+\frac{2}{x^2}+\frac{3}{x^3}dx.
Antiderivatives: Indefinite Integrals
If F(x)=ln⁡(2x2)F(x) = \ln(2x^2) is an antiderivative of f(x)f(x), find ff.
Practice: Definite Integral
Practice: Definite Integral
Evaluate ∫−20∣x2−1∣dx\displaystyle \int_{-2}^0\left|x^2-1\right|dx.
Find ∫(x23−x5+1x34)dx\displaystyle \int \left(\sqrt[3]{x^2}-\sqrt[5]{x}+\frac{1}{\sqrt[4]{x^3}}\right)dx
Practice: Indefinite Simplify First
Q:\textbf{Q:} Find ∫[x2−4xx+x(2x2−3x)]dx\displaystyle\int_{ }^{ }\left[\frac{x^2-4\sqrt{x}}{\sqrt{x}}+x\left(2x^2-3x\right)\right]dx
Practice: Indefinite Integral
Evaluate the integral ∫x+1+1x+2x2dx\displaystyle\int_{ }^{ }x+1+\frac{1}{x}+\frac{2}{x^2}dx
Practice: Indefinite Integral
Evaluate the integral ∫1+1x+2x2+3x3dx\displaystyle\int_{ }^{ }1+\frac{1}{x}+\frac{2}{x^2}+\frac{3}{x^3}dx.
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.
Antiderivatives and Indefinite Integrals
Find the most general antiderivative of f(x)=e2f\left(x\right)=e^2
Riemann Sums
Evaluate lim⁡n→∞ ∑i=1n[5n[(1+5in)2−3]]\displaystyle \lim_{n\to\infty}\ \sum_{i=1}^n\left[\frac{5}{n}\left[\left(1+\frac{5i}{n}\right)^2-3\right]\right]
Find ∫xe3 dx\int_{ }^{ }xe^3\ dx
Antiderivatives: Indefinite Integrals
If F(x)=ln⁡(2x2)F(x) = \ln(2x^2) is an antiderivative of f(x)f(x), find ff.
Antiderivatives: Indefinite Integrals
The graph of an antiderivative of f(x)f(x) is given below. Graph f(x)f(x).
Find an antiderivative for
f(x)=x5+2x4x\displaystyle f(x)=\frac{x^5+2\sqrt{x}}{4x}
Find an antiderivative for
f(x)=x5+2x4x\displaystyle f(x)=\frac{x^5+2\sqrt{x}}{4x}
Find ∫ln(e4x)dx\displaystyle \int ln(e^{4x})dx
If F(x)=ln⁡(2x2)F(x)=\ln(2x^2) is an antiderivative of f(x)f(x), find f.
Antiderivatives: Indefinite integrals
Evaluate the integral ∫1+1x+2x2+3x3dx\displaystyle\int_{ }^{ }1+\frac{1}{x}+\frac{2}{x^2}+\frac{3}{x^3}dx.
Practice: Indefinite Simplify First
Q:\textbf{Q:} Find ∫[x2−4xx+x(2x2−3x)]dx\displaystyle\int_{ }^{ }\left[\frac{x^2-4\sqrt{x}}{\sqrt{x}}+x\left(2x^2-3x\right)\right]dx
Find ∫(x23−x5+1x34)dx\displaystyle \int \left(\sqrt[3]{x^2}-\sqrt[5]{x}+\frac{1}{\sqrt[4]{x^3}}\right)dx
Antiderivatives: Indefinite integrals
Evaluate the indefinite integral ∫x23−x5+1x34dx\displaystyle \int \sqrt[3]{x^2}-\sqrt[5]{x}+\frac{1}{\sqrt[4]{x^3}}dx.
Indefinite Integrals: Antiderivatives
Find the indefinite integral ∫2x(2x+3)2dx\int2x(2x+3)^2dx
Find ∫xe3 dx\int_{ }^{ }xe^3\ dx
Antiderivatives: Indefinite Integrals
Find ∫[x2−4xx+x(2x2−3x)]dx\int_{ }^{ }\left[\frac{x^2-4\sqrt{x}}{\sqrt{x}}+x\left(2x^2-3x\right)\right]dx.
Practice: Definite Integral
Practice: Definite Integral
Evaluate ∫−20∣x2−1∣dx\displaystyle \int_{-2}^0\left|x^2-1\right|dx.
Antiderivatives: Indefinite Integrals
For the following function, find the most general antiderivative.
f(x)=x5+2x4x\displaystyle f(x)=\frac{x^5+2\sqrt{x}}{4x}
Find the antiderivative FF
f(x)=4x+x22+5,F(1)=1/3\displaystyle f(x)=4x+\frac{x^2}{2}+5, F(1)=1/3