Math ↠ Definite integral ↠ Calculating indefinite integrals

Calculating indefinite integrals

A indefinite integral can be calculated with the fundamental theorem of calculus by inserting and subtracting the limits of integration $a$ and $b$ into an antiderivative of $f$:

$\int_a^b f(x) \, \mathrm{d}x$ $= [F(x) + C]_a^b$ $= F(b) - F(a)$
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Hint

In order to be able to find the antiderivative, one should master the integration rules.
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Method

  1. Calculate antiderivative
  2. Insert limits of integration into antiderivative
  3. Calculate integral: $F(b)-F(a)$

Example

$\int_\color{red}{2}^\color{blue}{3} 3x^2 \, \mathrm{d}x$

  1. Calculate antiderivative

    Here the power rule is applied.
    $F(x)=\int 3x^2=x^3$

  2. Insert limits of integration into antiderivative

    Now $x$ of the antiderivative is replaced with the limits of integration of the integral.
    $F(\color{red}{a})=F(\color{red}{2})=\color{red}{2}^3$
    $F(\color{blue}{b})=F(\color{blue}{3})=\color{blue}{3}^3$

  3. Calculate integral

    Now this only has to be inserted in the formula.
    $\int_\color{red}{a}^\color{blue}{b} f(x) \, \mathrm{d}x$ $= [F(x) + C]_\color{red}{a}^\color{blue}{b}$ $= F(\color{blue}{b}) - F(\color{red}{a})$

    $\int_\color{red}{2}^\color{blue}{3} 3x^2 \, \mathrm{d}x$ $= [x^3]_\color{red}{2}^\color{blue}{3}$ $= \color{blue}{3}^3 - \color{red}{2}^3$ $=27-8$ $=19$