Integrals

Antiderivatives and constants

Reverse differentiation. An indefinite integral represents a family of functions, not one number.

1. A family of primitives

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Example graph

F(x)=x^2+C,\quad F\prime(x)=2x

Every curve x²+C has derivative 2x. Moving C shifts the primitive vertically without changing any of its slopes. A known value F(0) selects one member.

C: 0
F(0): 0

The formulas identify the solid and dashed curves. Compare their values using the slider; the accompanying explanation states their mathematical relationship.

2. Integrate a power

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Example graph

F(x)=\frac{x^3}{3}+C,\quad F\prime(x)=x^2

For n≠−1, ∫xⁿ dx=xⁿ⁺¹/(n+1)+C on an interval where the powers are defined. For x², increase the exponent to 3 and divide by 3.

C: 0
F(0): 0

The formulas identify the solid and dashed curves. Compare their values using the slider; the accompanying explanation states their mathematical relationship.

3. The exceptional exponent

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Example graph

F(x)=\ln|x|+C,\quad F\prime(x)=\frac1x

The power formula cannot divide by n+1 when n=−1. Instead, ∫1/x dx=ln|x|+C on intervals excluding zero. Constants on the two separate half-lines need not match.

C: 0
F(1): 0

The formulas identify the solid and dashed curves. Compare their values using the slider; the accompanying explanation states their mathematical relationship.

4. An exponential primitive

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Example graph

F(x)=e^x+C,\quad F\prime(x)=e^x

Since the derivative of eˣ is eˣ, its primitives are eˣ+C. For e^(kx) with nonzero constant k, divide by k to undo the chain-rule factor.

C: 0
F(0): 1

The formulas identify the solid and dashed curves. Compare their values using the slider; the accompanying explanation states their mathematical relationship.