Basics of Evaluating Integrals

In a Nut Shell: The key strategy in evaluating any complicated integral is to simplify it

into one or more standard integrals where you know each standard integral.

Therefore you must memorize the Standard Form Integrals - - This is mandatory.

Standard integrals include integrals involving functions such as polynomials,

exponentials, trig functions, combinations of these and others. You must also

memorize trig identities. Click here for them.

Listed below are methods, procedures, and transformations commonly used to simplify

complicated integrals into ones with standard form.

Note: You may apply them separately, in combination, or group terms.

1.	Simple substitution
2.	Use of trig formulas
3.	Combination of substitution and trig formulas
4.	Grouping of terms along with a substitution
5.	Substitution followed by grouping followed by another substitution
6.	Integration by parts
7.	Combination of substitution and integration by parts
8.	Any of the above in combination perhaps multiple times
9.	Manipulation
10.	Intuition using knowledge of derivative

Use of a simple substitution I = ∫(x + 1)² dx let u = x + 1, du = dx

Then the integral becomes ∫ u² du which is of the standard form

∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c Result: I = ( x + 1 )³ / 3 + C

Use of Trig formulas (in this case twice) Trig formula sin²x = (1 – cos 2x)/2

cos²x = (1 + cos 2x)/2

Example: I = ∫ sin⁴x dx = ∫ [ (1 – cos 2x)/2 ]² dx

I = ¼ ∫ [ (1 – 2cos 2x + cos² 2x) dx

I = ¼ ∫ [ (1 – 2cos 2x + ( 1 + cos 4x )/2 ]dx

So I = ¼ ∫ [ 1 – 2cos 2x ] dx + 1/8 ∫ ( 1 + cos 4x ) dx which are standard integrals

Result: I = (1/8) x + (1/32) sin 4x + C

Click here to continue with examples of key types.

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