Integration Methods Flow Chart.

Remember with any mathematical problem look to see if we can simplify before proceeding with any calculation or algebra. If you have an idea what the integration is then check you possible solution by differentiation. If it's not the same as the integrand can it be multiplied by a constant to be correct?

If inspection doesn't work then work through the following.

To solve a given integration problem, firstly, simplify the integrand using one or more of these:

Remove any constant factors from the integrand.

$$\int \mathrm{a}f\left(x\right)dx=\mathrm{a}\int f\left(x\right)dx$$

Split the integrand term-wise.

$$\int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx$$

Secondly, follow this flow chart.

Is the integrand of a standard form?

 ⇨ YES, then use the standard integrals from memory or from the formulae booklet.

 ⇩
NO.

Is the integrand of the form

$$\frac{f^{\prime }\left(x\right)}{f\left(x\right)}$$

 ⇨ YES, then

$$\int \frac{f^{\prime }\left(x\right)}{f\left(x\right)}dx=ln\left|f\left(x\right)\right|$$

 ⇩
NO.

Is the integrand of the form

$$f^{\prime }\left(x\right){\left[f\left(x\right)\right]}^n$$

 ⇨ YES, then

$$\int f^{\prime }\left(x\right){\left[f\left(x\right)\right]}^{n}dx=\frac{{\left[f\left(x\right)\right]}^{n+1}}{n+1},n\ne -1$$

 ⇩
NO.

Is the integrand a bottom heavy proper fraction?

 ⇨ YES, then use partial fractions and integrate term-wise (see above).

 ⇩
NO.

Is the integrand a product of two distinct functions?

 ⇨ YES, then use integration by parts (formula in the booklet).

 ⇩
NO.

Then either integration by substitution or simplify the integrand, perhaps with a trig identity .

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