Indefinite Integral Rules (College Board AP® Calculus AB)

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When integrating sums or differences of terms
- the integral is simply the sum (or difference) of the integrals of the terms
- This may be expressed as $\int (f (x) \pm g (x)) d x = \int f (x) d x \pm \int g (x) d x$
  - E.g. $\int (x^{2} + \cos x) d x = \int x^{2} d x + \int \cos x d x = \frac{1}{3} x^{3} + \sin x + C$
  - We still only need a single constant of integration
- Note that products and quotients of terms cannot be integrated in this way
  - They may need to be expanded or simplified first
  - Or a more advanced integration technique may need to be used
    - See the 'Methods of Integration' study guide
When integrating a constant multiple of a term
- the term may be brought out in front of the integral as a multiplier
- This may be expressed as $\int k f (x) d x = k \int f (x) d x$
  - E.g. $\int 5 e^{x} d x = 5 \int e^{x} d x = 5 e^{x} + C$
The rules for sums, differences and constant multiples can be combined
- If $p$ and $q$ are constants, then
  - $\int (p f (x) \pm q g (x)) d x = p \int f (x) d x \pm q \int g (x) d x$
  - This idea can be extended for any number of terms inside the integral

Find the indefinite integral $\int (3 x^{3} - 2 \sin 3 x + 5 e^{4 x}) d x$ .

Answer:

Write as a sum or difference of integrals, with constants pulled in front as multipliers

$3 \int x^{3} d x - 2 \int \sin 3 x d x + 5 \int e^{4 x} d x$

Once you've become confident with integration, you can skip this step

Now integrate the terms one by one

$3 \cdot \frac{1}{4} x^{4} - 2 \cdot (- \frac{1}{3} \cos 3 x) + 5 \cdot \frac{1}{4} e^{4 x} + C$

Don't forget the constant of integration

Multiply out and simplify

$\frac{3}{4} x^{4} + \frac{2}{3} \cos 3 x + \frac{5}{4} e^{4 x} + C$

the (exam) results speak for themselves:

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