Integration by Substitution
Integration by substitution (as seen in SL 5.10) simplifies complex integrals into a form with which we can integrate.
The general method can be seen as.
$$\int kf(g(x))g'(x)dx$$
where we make $u = g(x)$ such that $du = g'(x)dx$ and $dx = \frac{du}{g'(x)}$.
Therefore the integral becomes
$$k\int f(u)g'(x)\,\frac{du}{g'(x)} = k \int f(u)\,du$$
where $k$ is a constant.
Of course we would not want to do this if $f(x)$ is a function that's even more difficult to integrate since it is a method to simplify! Choose $u=g(x)$ wisely.
NoteIn IB examinations, if the integral is not in the form $\int kg'(x)f(g(x))dx$, the substitution will be provided.
Steps for Integration by Substitution:
- Identify a suitable—suitable based on your judgement—substitution $u = g(x)$
- Calculate $ \frac{du}{dx} = g'(x)$
- Express the integral in terms of $u$
- Integrate with respect to $u$
- Substitute back to express the result in terms of $x$
Let's integrate $\int x\sqrt{1-x^2}dx$
- Let $u = 1-x^2$
- $du = -2xdx$ or $-\frac{1}{2}du = xdx$
- The integral becomes $-\frac{1}{2}\int \sqrt{u}du$
- Integrating: $-\frac{1}{2} \cdot \frac{2}{3}u^{3/2} + C = -\frac{1}{3}u^{3/2} + C$
- Substituting back: $-\frac{1}{3}(1-x^2)^{3/2} + C$
Look for parts of the integrand that appear both as a function and its derivative. These are often good candidates for substitution.
TipRemember that for a definite integral in the form
$$\int^a_b f(x) dx$$
when doing the substitution $u = g(x)$, it is necessary to change the bounds of the integral since we're now integrating with respecting to $u$ and not $x$.
So the integral becomes:
$$\int^{g(a)}_{g(b)} f(g^{-1}(u))g'(g^{-1}(u)) du$$
Which will hopefully be easier than whatever you were doing before.
Integration by substitution can often be tricky because it can easily make an integrand more complicated if you choose the wrong substitution. Make sure to think about what the substitution will do before you choosing it, and don't be afraid to start over.
Do not try to memorise every substitution for every integral. They seem stupid to memorise—and they are. You'll learn these work with enough practice. Eventually you'll be able to come up with your own substitutions seemingly out of thin air.
Integration by Parts
Integration by parts is a method used when the integrand is a product of two functions. It's based on the product rule of differentiation.
