Integration by Change of variable

Integration by Change of variable . This method consists of transforming the given integral into a simpler one by changing the independent variable .

Summary

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• 1 Definition
• 2 Examples
• 3 Sources
• 4 See also

Definition

There are several integration methods , all of them consisting of reducing the sought integral to an already known integral, such as one of the ones in the table, or reducing it to a simpler integral.

The method of integration by substitution or change of variable is based on the derivative of the compound function.

We want to perform the integral ∫ ƒ (x) dx where ƒ does not have an immediate primitive. We must look for a change of variable that transforms the integral into an immediate integral or composition of functions. So for the change,

x = g (t)

dx = g ′ (t) dt

∫ ƒ (x) dx = ∫ f (g (t)) g ′ (t) dt

In this way the integrand has been transformed according to the new variable t . If the choice of the variable t has been successful, the resulting integral is easier to integrate. The success of integration depends, to a considerable degree, on the ability to choose the appropriate substitution of the variable.

Once the primitive function, F (t) + C , is obtained, the change of the variable is undone by substituting t = g (x) .

Thus we have the indefinite integral as a function of the initial variable x .

Examples

1. Find ∫ (3x – 5) ⁴dx

Solution. In this simple case we can see that this integral “looks like”

⁴ u ⁴ du , which suggests us to take the change of variable u = 3x-5 .

u = 3x-5 ⇒ du = 3 dx ⇒ dx = (1/3) du

Substituting in the integral,

∫ (3 x – 5) ⁴ dx == ∫u ⁴ du / 3 = ⅓ ∫ u ⁴ du

= ⅓ (or ⁵ /5) + u c = ⁵ /15 + c = ∫ (3 x – 5) ⁵ /15 + c

2. Find ∫ cos ⁴x sinx dx

Solution. In this case the integral “looks like”

∫ u ⁴ du

which suggests us to take the change of variable u = cosx .

u = cosx ⇒ du = -senx dx ⇒ dx = -du

∫ (cos) 4 (sinx dx) = ∫ (u ⁴ ) (-du) = – ∫u ⁴ du = – (or ⁵ /5) + c

= – (Cos ⁵ x / 5) + c