Evaluate the integrals using u-substitution:

Question

Evaluate the integrals using u-substitution:

∫r^2((r^3/18)-1)^5 dr

Answer 1

The given integral is:

∫r²((r³/18) + 1)⁵dr

Looking at the different parts of the integral, we notice there is one part that contains the term r² and one that contains the term r³. When you see this, this is generally a good sign that u-sub can be used since taking the derivative of whatever r³ term there is will leave you with a r² that can be used to cancel out the original r² term.

Trying this, we can say that:

u = (r³/18) + 1

Taking the derivative of this leaves us with:

du/dr = r²/6 (application of the power rule)

From this, we want to solve for dr since dr is found in the original integral:

du/dr = r²/6

dr = 6du/r²

Now that we have found dr, we can substitute both this term and our u = (r³/18) + 1 back into our original integral:

∫r²((r³/18) + 1)⁵dr (original integral)

∫r²(u)⁵(6du/r²)

Simplifying, we get:

∫(u)⁵(6du)

We can then take out the "6" using the Constant Rule:

∫(u)⁵(6du)

6 ∫u⁵du

From here, the integral is much simpler to solve:

6 ∫u⁵du

6 (1/6 u⁶ + C)

Simplifying, we get:

u⁶ + C

However, we must now substitute back in the original substitution we made which was u = (r³/18) + 1. Doing so we get:

((r³/18) + 1)⁶ + C