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Revision:Integration by Substitution

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Integration by Substitution

It is possible to transform a difficult integral to an easier integral in a different variable using a substitution. By using substitutions, we can show that:


\displaystyle \int_{x_1}^{x_2}f(x)\text{d}x = \int_{u_1}^{u_2}g(u)\text{d}u where u = h(x) and g(u) is much easier to integrate than f(x). Here, \frac{\text{d}u}{\text{d}x} = h'(x) and so \text{d}x=\frac{\text{d}u}{h'(x)}


Worked Examples

Find \displaystyle \int x(x^2+1)^5\text{d}x Here, we have a rather nasty x^2+1 term inside the bracket, so we let u=x^2+1.

We can now find \frac{\text{d}u}{\text{d}x}=2x by simple differentiation. By rearrangement, \text{d}x=\frac{\text{d}u}{2x} Making the u-substitution and the dx-substitution in the original problem, we transform it to \displaystyle \int \frac{x u^5}{2x}\text{d}u = \int \frac{u^5}{2}\text{d}u = \frac{u^6}{12} + c

And since u=x^2+1, \displaystyle \int x(x^2+1)^5\text{d}x = \frac{(x^2+1)^6}{12} + c

Using a Substitution

Sometimes you will be told to integrate a function by using a substitution. For example, in the above first example, you might be told u=x^2+1 without being having to make it up yourself.

Comments

Diagrams are missing here. Alternatively, the necessary maths could be written in LaTeX.

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