bbroocks said:Homework Statement
∫(4x^3)/√(x^2+4)dx
Homework Equations
The Attempt at a Solution
So, I let x= 2tanθ
dx= 2sec^2θ dθ
So, √(4tan^2(θ)+4)=2secθ
∫(4x^3)/√(x^2+4)dx=∫((32tan^3(θ))/(2secθ))2sec^2(θ)dθ.
Would it go to ∫16tan^3(θ)2sec(θ)dθ
or ∫32tan^3(θ)sec(θ)dθ
Trig substitution is a technique used in integration to simplify and evaluate integrals that involve functions with trigonometric identities.
Trig substitution is used to transform integrals involving algebraic or radical expressions into integrals involving trigonometric functions, which can be easier to evaluate.
The appropriate trig substitution is chosen based on the form of the integral. Common substitutions include substitution for expressions involving x^2 + a^2, x^2 - a^2, and a^2 - x^2.
The trigonometric identity used in trig substitution for ∫(4x^3)/√(x^2+4) is x = 2tanθ. This substitution transforms the integral into ∫8tan^3θsecθdθ, which can be simplified and evaluated.
After applying trig substitution, the integral can be simplified using trigonometric identities and then evaluated using integration techniques such as integration by parts or partial fractions.