Integration by Substitution

Integrals using substitution

Open Ended

If u= $$x^2$$  and f(u)= $$u^3$$    then find $$\int_{ }^{ }f\left(u\right)du$$  as a function of u.

Open Ended

If u= $$x^2$$   and f(u)= $$u^3$$ then find $$\int_{ }^{ }f\left(u\right)du\$$  as a function of x this time.  

Multiple Choice

If u=x^2, what does du = ?

• du=dx

• du=2x dx

• $$\frac{du}{dx}=2x$$

• du= $$\frac{1}{3}x^{3\ }$$

• du= $$x^3+c$$

Open Ended

$$\int_{ }^{ }f\left(x\right)dx\ \ \ \ \ \ \int_{ }^{ }f\left(u\right)du\ \ \ \ \ \ \int_{ }^{ }f\left(x\right)du\ \ \ \ \ \int_{ }^{ }f\left(u\right)dx$$  

Let f(x)=x3+1 and u=x2, which of these are equivalent?

Integration using substitution

Identify an "inside" function as your u-function

Then find the derivative of that u-function in terms of x.

Now you can simplify and integrate

Substitute the u-function and the value of du to make the integrand easier to integrate, then integrate and finally substitute back to have an equation in x.

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