Integration by Change of variable . This method consists of transforming the given integral into a simpler one by changing the independent variable .
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- 1 Definition
- 2 Examples
- 3 Sources
- 4 See also
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 .
- 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
- 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