Integration Techniques (Reduction & Trig Substitution)

[ChaoticLlama]

Could I please have detailed explanations for the following two problems. I missed the classes in which these techniques were taught and do not understand them completely.

Reduction:
∫e^(5*x) * cos(3x)^2

Trig Substitution:
∫x²dx / √(x² - 1)

thanks for your time

 

[askmemath]

Trig Substitution:
∫x²dx / √(x² - 1)

I hope you know the trig identities
sec^2 t = 1 + tan^2 t
cot^2(x) + 1 = csc^2(x)

Substitute in place of X .

Come back with a re-post if you get stuck.

 

[ChaoticLlama]

∫x²dx / √(x² - 1)

let x = tan[θ]
dx = sec[θ]^2dθ

∫(tan[θ]^2 * sec[θ]^2)dθ / tan[θ]
∫(tan[θ] * sec[θ]^2)dθ

Once I get here i'm not sure when to integrate the trig functions.

 

[soroban]

Hello, ChaoticLlama!

You did okay up to a point . . .

∫x²dx / √(x² - 1)

let x = tan θ
dx = sec² θ dθ

The radical becomes: .tan θ

. . . . . . . . . . . . . . . . . . . .sec²θ
The integral becomes: . --------- (sec θ tan θ dθ) . = . sec³θ dθ
. . . . . . . . . . . . . . . . . . . .tan θ

The dreaded "secant-cubed" requires a repeated by-parts solution.

. . . The answer is: . (1/2) [sec θ tan θ + ln|sec θ + tan θ|] + C


And I'll let you back-substitute . . .