Question

Evaluate the integral by making the given substitution.

the integral of cos^6 θ sin θ dθ, u = cos θ

Answer 1

So with a substitution, we want to replace ALL of the Theta's and D(Theta) with U's and du.

Let \[\large u=\cos \theta\]Taking the derivative gives us,\[\large du=-\sin \theta d \theta\]
But our equation includes a sin theta, without the negative, so let's move the negative to the other side of that equation.
\[\large -du= \sin \theta d \theta\]

Answer 2

So now we can replace the cos theta with something involving u, and we can replace the sin theta d(theta) with something involving du, and it should simplify our integral.

Answer 3

\[\large \int\limits\limits (\cos \theta)^6 (\sin \theta d \theta) \rightarrow \quad \int\limits\limits (u)^6(-du)\]

Answer 4

Understand what we did there? :D Substitutions can be a little tricky to get used to. It's not as easy as differentiation was ^^

Answer 5

ohh okay i see what i did wrong now. thank you!