Techniques of Integration

Integrating f(ax+b)

SL 5.10

If F(x)=∫f(x)dx, then

∫f(ax+b)dx=a1F(ax+b)🚫

Integration by substitution

SL 5.10

Integrating a composition of functions f(g(x)) requires us to divide by g′(x), so it is easier to find the anti-derivative of anything of the form g′(x)f′(g(x)) by first dividing by g′(x).

In symbols, we use the known fact

∫kg′(x)f′(g(x))dx=kf(g(x))+C🚫

and let u=g(x), giving us

∫kg′(x)f′(g(x))dx=k∫f′(u)du,🚫

an integral we can solve more easily:

k∫f′(u)du=kf(u)+C.

Then, we substitute g(x) back in to get our desired result of kf(g(x))+C.

Substitution and Integral Bounds

SL 5.10

When we make a substitution in a definite integral in the form

∫abkg′(x)f′(g(x))dx

we need to remember that the bounds are from x=a to x=b:

∫abkg′(x)f′(g(x))dx =k∫x=ax=bf′(u)du =[kf(u)]x=ax=b

We then have two choices:

Plug x=a and x=b into u to find the bounds in terms of u.
Plug u(x) back in and use the bounds a→b.

Exercises

Key Skills

Exercises

Key Skills

Exercises

Key Skills

Integrating f(ax+b)

Integration by substitution

Substitution and Integral Bounds

Integrating f(ax+b)

Integration by substitution

Substitution and Integral Bounds