Calculus Solver Integral of 2x
The integral of 2x with respect to x is x² + C. The power rule for integration raises the exponent by one and divides by the new exponent, and the constant C is there because differentiating any constant gives zero, so infinitely many functions share this derivative.
Using the power rule
The power rule for integration says ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C, for any n other than -1. Here 2x is 2x¹, so n = 1.
Pull the constant out and apply it: ∫2x dx = 2 ∫x dx = 2 · x²/2 + C = x² + C. The 2 you factored out and the 2 from the new exponent cancel, which is why this particular integral comes out so clean.
Why the + C
Differentiate x², x² + 7 and x² - 400 and all three give 2x, because the derivative of a constant is zero. Going backwards, there is no way to know which constant was there.
So the honest answer is the whole family, written x² + C. Leaving off the C is the most commonly deducted mark in an integration exam, and it is not pedantry: the family is the answer.
Between two limits
A definite integral drops the C, because it cancels: evaluate the antiderivative at the top limit and subtract its value at the bottom.
∫ from 0 to 3 of 2x dx = [x²] from 0 to 3 = 9 - 0 = 9. The diagram checks it: the region is a triangle of base 3 and height 6, area ½ · 3 · 6 = 9.
Nearby integrals, by the same rule
| Integral | Answer |
|---|---|
| ∫2x dx | x² + C |
| ∫x dx | x²/2 + C |
| ∫3x² dx | x³ + C |
| ∫2 dx | 2x + C |
| ∫(1/x) dx | ln|x| + C, the exception the power rule cannot handle |
Questions
What is the integral of 2x?
x² + C. Differentiating x² + C returns 2x, which is the check.
What is the integral of 2x from 0 to 3?
9. Evaluate x² at 3 and at 0 and subtract: 9 - 0 = 9. The constant cancels in a definite integral.
Why is there a plus C?
Because every function of the form x² + C has derivative 2x. Without more information the constant cannot be pinned down, so the answer is the whole family.
More calculus reference
Stuck on the actual problem
Photograph it and get the worked solution, step by step, with the rule that justified each line named.
