calculaus u sub

Question

∫6(tanx)⁵(secx)²dx

Answer 1

Problem:

∫6sec2(x)tan5(x)dx

Substitute u=tan(x) ⟶ du/dx=sec2(x) ⟶ dx=sec2(x)du:

=6∫u5du

Now solving:

∫u5du

Apply power rule:

∫undu=un+1n+1 with n=5:

=u66

Plug in solved integrals:

6∫u5du

=u6

Undo substitution u=tan(x):

=tan6(x)

∫6sec2(x)tan5(x)dx

=tan6(x)+C

Answer 2

Great question McKenzie! Let me make this simple!

Notice that the derivative of tanx is equal to (secx)^2, and the integral has both tanx and (secx)^2.

Therefore, let u = tanx, and by taking the derivatives on both sides, du/dx = (secx)^2 ⇒ du = (secx)^2dx

6 * ∫ (u ^ 5)du = u^6 + C = (tanx) ^ 6 + C >>>>>>>>>>>>>>>>>>> (ANSWER) Hope this helped you!