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Integration: reversing differentiation

Integration is the reverse of differentiation. If you know the gradient function, integration gets you back to the curve, up to a constant.

The rule

∫ xn dx = xn+1/(n + 1) + c, for any rational n except n = -1.

Add 1 to the power, then divide by the new power. Integrate term by term.

The constant c

Differentiating removes constants, so every indefinite integral needs + c. Rewrite roots and fractions as powers first, exactly as for differentiation.

Finding the curve

If you are given dy/dx and a point on the curve, integrate to get y with + c, then substitute the point to find c.

Worked example

dy/dx = 6x2 - 4x and the curve passes through (1, 5). Find y.

  1. y = 2x3 - 2x2 + c
  2. At (1, 5): 5 = 2 - 2 + c
  3. c = 5

Answer: y = 2x3 - 2x2 + 5

Key idea: ∫ xn dx = xn+1/(n + 1) + c (n not -1). Use a known point to find c.

Check yourself

  1. Find ∫ 4x3 dx.
    • 12x2 + c
    • x4 + c
    • 4x4 + c
    • x4
    Show answer

    x4 + c. 4 x x4/4 = x4, plus the constant.

  2. Find ∫ x-2 dx.
    • x-1 + c
    • -2x-3 + c
    • -x-3/3 + c
    • -x-1 + c
    Show answer

    -x-1 + c. x-1/(-1) = -x-1.

  3. Find ∫ 3sqrt(x) dx.
    • (3/2)x-1/2 + c
    • 2x3/2 + c
    • 3x3/2 + c
    • (9/2)x3/2 + c
    Show answer

    2x3/2 + c. 3x1/2 integrates to 3 x x3/2/(3/2) = 2x3/2.

  4. f'(x) = 2x + 3 and f(0) = 4. Find f(2).
    Show answer

    14. f(x) = x2 + 3x + c, and f(0) = 4 gives c = 4. f(2) = 4 + 6 + 4 = 14.

  5. dy/dx = 3x2 - 2 and the curve passes through (2, 10). Find the constant c in y.
    Show answer

    6. y = x3 - 2x + c. 10 = 8 - 4 + c, so c = 6.

  6. dy/dx = 8/x3 and the curve passes through (1, 1). Find y when x = 2.
    Show answer

    4. 8x-3 integrates to -4x-2 + c. At (1, 1): 1 = -4 + c, so c = 5. At x = 2: y = -4/4 + 5 = 4.

  7. Why does the rule ∫ xn dx = xn+1/(n + 1) + c not work for n = -1?
    • It would mean dividing by n + 1 = 0
    • x-1 cannot be differentiated
    • Because x0 = 1
    • Because the answer would be negative
    Show answer

    It would mean dividing by n + 1 = 0. With n = -1 the new power is 0 and you would divide by 0.

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Last updated 7 November 2026. Spotted a mistake? Tell us.