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Wize AP Calculus (BC) Textbook > Integration & Accumulation of Change

Antiderivatives & The Indefinite Integrals

Computing Definite Integrals GeometricallyPrevious Section
Antiderivatives of Trig FunctionsNext Section
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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