Integrals Using Substitution and Integration by Parts

In a Nut Shell: In some integrals you may combine one or more substitutions along

with one or more integration by parts (in either order). Again the objective is to

transform the original integral into one or more standard integrals.

Here’s an example Combining a Substitution with Integration by Parts

Substitution and I = ∫ u dv = uv - ∫ v du

Example: I = ∫( x³ /√(1 - x² )dx

Substitution: x = sin θ dx = cos θ dθ

now 1 - x² = 1 – sin² θ = cos² θ

So I = ∫ (sin³ θ cos θ / cos θ ) dθ = ∫ sin³ θ dθ

Write as ∫ sin² θ sin θ dθ and now integrate by parts

let u = sin² θ, dv = sin θ d θ <-- Standard integral

du = 2 sin θ cos θ dθ v = - cos θ

Then I becomes -sin² θ cos θ + 2∫sin θcos² θ dθ

Now let w = cos θ, dw = - sin θ dθ and integral becomes

one in standard form:

I = -sin² θ cos θ - 2∫w² dw = -sin² θ cos θ - (2/3) w³ + c

Or I = -sin² θ cos θ - (2/3)cos³ θ + c

Finally, express result in terms of original variable x

Recall sin θ = x then cos θ = √(1 – x²) so

Result: I = -x² √(1 – x²) - (2/3) (1 – x²)^3/2 + c

Return to Notes for Calculus 2

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