U substitution on indefinite integral/antiderivative.
(t^3+4)^-2 t^2
On this should I use u on the parenthesis and should I put this into a natural log?
if you let
u = t^3+4 then
du = 3t^2 dt
Now your integrand becomes
u^-2 (du/3) = 1/3 u^-2 du
that should be a cinch.
Yes, had just finished getting it! Thanks!
To solve the integral of (t^3+4)^-2 t^2 using a u-substitution, you can indeed substitute u for the expression inside the parenthesis, t^3 + 4. However, using a natural logarithm is not necessary in this case. Let me explain the steps in detail:
Step 1: Choose u
For a u-substitution, let's choose u as the expression inside the parenthesis: u = t^3 + 4.
Step 2: Find du/dt and solve for dt
Now, calculate the derivative of u with respect to t: du/dt = 3t^2. To solve for dt, divide both sides by 3t^2: dt = du / (3t^2).
Step 3: Substitute u and dt in the integral
Now, substitute u = t^3 + 4 and dt = du / (3t^2) into the integral:
∫ (t^3+4)^-2 t^2 dt = ∫ u^-2 t^2 (du / (3t^2)).
Step 4: Simplify the integral
Simplify the expression: ∫ u^-2 du / 3.
Step 5: Integrate
Now, integrate the simplified expression: (1/3) ∫ u^-2 du.
Step 6: Evaluate the integral
Evaluate the integral: (1/3) ∫ u^-2 du = (1/3) (-u^-1) + C = -1/(3u) + C.
Step 7: Replace u with the original expression in terms of t
Replace u with the original expression, t^3 + 4:
-1/(3(t^3 + 4)) + C.
Therefore, the indefinite integral of (t^3+4)^-2 t^2 is -1/(3(t^3 + 4)) + C.