Integration using Methods of Substitution – Click here for applications to Arc Length

In a Nut Shell: The general strategy to evaluate an integral with a complicated integrand is

to transform the complicated integral (not in standard form) into a form identical to one or more

of the standard form integrals using simple substitutions. You then apply your knowledge of

the standard integrals to obtain the value of the original integral.

The nice thing about evaluating integrals is that you can always check your result by taking

the derivative of your result. It should be the same as the integrand in the original integral.

Example 1: I = ∫(x + 1)² dx try the substitution u = x + 1,

Then du = dx

So the integral becomes ∫u² du which has the standard form

∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c

In this case x = u so that the integral is ∫u ² dx and I = u³ / 3 + c

Finally you need to express your result in terms of the original variable x .

So replace u with x + 1 to obtain I = (x + 1) ³ / 3 + c

where c is the constant of integration.

Check: dI/dx = 3 [(x + 1)² ] / 3 = (x + 1)² which is the integrand

Note: A second option is to first multiply out (x + 1)² to obtain x² + 2x + 1 as the

integrand. Then evaluate each "standard" integral ∫ x² dx + ∫ 2x dx + ∫ (1) dx.

Example 2: I = ∫ cos x sin x dx

Try the substitution u = sin x, so du = cos x d x

Then the integral I becomes a standard integral I = ∫ u du = u² / 2 + c

Finally express your result in terms of the original variable x .

So replace u with sin x to obtain

I = sin² x + c

Check: dI/dx = [ 2 sin x cox x ] / 2 = sin x cox x which is the integrand

Return to Notes for Calculus 1

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