VG2
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Quiz
What is the indefinite integral?
The indefinite integral is the reverse of differentiation — it finds the antiderivative F(x).
Differentiation and integration are opposite operations. While differentiation finds the rate of change, integration finds the original function.
∫ f(x) dx = F(x) + C
F(x) is called the antiderivative of f(x). C is the constant of integration — representing all possible vertical shifts of the curve.
Key integration rules
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)
∫ k dx = kx + C
∫ eˣ dx = eˣ + C
∫ 1/x dx = ln|x| + C
∫ cos(x) dx = sin(x) + C
∫ sin(x) dx = -cos(x) + C
∫ k dx = kx + C
∫ eˣ dx = eˣ + C
∫ 1/x dx = ln|x| + C
∫ cos(x) dx = sin(x) + C
∫ sin(x) dx = -cos(x) + C
Examples
Example 1: Find ∫ 3x² dx
∫ 3x² dx = 3 · x³/3 + C = x³ + C
Example 2: Find ∫ (2x + 5) dx
∫ (2x + 5) dx = x² + 5x + C
Always remember + C! Without the constant of integration the answer is incomplete — there are infinitely many antiderivatives differing by a constant.
The infinite! No other question has ever moved so profoundly the spirit of man.
— David Hilbert (1862–1943)
🧠 Test yourself
Question 1 of 5
What is ∫ x³ dx?
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