Find the indefinite integral ∫(6x^2 - 4x + 5) dx.
Solution
Problem
Find the indefinite integral ∫(6x^2 - 4x + 5) dx.
Step 1: Integrate each term separately.
We need to evaluate:
∫(6x^2 - 4x + 5) dx
Use the power rule for integration:
∫x^n dx = x^(n+1)/(n+1) + C, for n ≠ -1
Step 2: Integrate 6x^2.
∫6x^2 dx = 6(x^3/3)
Simplify:
6(x^3/3) = 2x^3
Step 3: Integrate -4x.
∫-4x dx = -4(x^2/2)
Simplify:
-4(x^2/2) = -2x^2
Step 4: Integrate 5.
∫5 dx = 5x
Step 5: Combine the results and include the constant of integration.
∫(6x^2 - 4x + 5) dx = 2x^3 - 2x^2 + 5x + C
The constant C is necessary because differentiating any constant gives zero.
Therefore, the indefinite integral is 2x^3 - 2x^2 + 5x + C.
Result
2x^3 - 2x^2 + 5x + C