Integration Techniques (Reduction & Trig Substitution)

ChaoticLlama

Junior Member
Joined
Dec 11, 2004
Messages
199
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

New member
Joined
Apr 28, 2005
Messages
31
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

Junior Member
Joined
Dec 11, 2004
Messages
199
∫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

Elite Member
Joined
Jan 28, 2005
Messages
5,588
Hello, ChaoticLlama!

You did okay up to a point . . .

x²dx / √(x² - 1)

let x = tan θ
dx = sec<sup>2</sup> θ dθ
The radical becomes: .tan θ

. . . . . . . . . . . . . . . . . . . .sec<sup>2</sup>θ
The integral becomes: . --------- (sec θ tan θ dθ) . = . sec<sup>3</sup>θ 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 . . .
 
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