Free Pearson Edexcel International A-level Mathematics · P1

Indefinite integration and finding a curve

What you’ll learn

  • Integrate sums of powers, excluding exponent −1.
  • Find the constant of integration from a point on the curve.

Understand the idea

Integration reverses differentiation. For a power, increase the exponent by one and divide by the new exponent. This rule excludes x−1x^{-1} because the resulting denominator would be zero. Rearrange a quotient into individual powers before integrating.

An indefinite integral describes a family of curves that differ by a vertical translation. The constant CC records that freedom. When a point is supplied, substitute its coordinates into the integrated expression to find the single member of the family. Check the result by differentiating it.

Key formulae

∫xn dx=xn+1n+1+C(n≠−1)\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\quad(n\ne-1)

∫[af(x)+bg(x)] dx=a∫f(x) dx+b∫g(x) dx\int[af(x)+bg(x)]\,dx=a\int f(x)\,dx+b\int g(x)\,dx

y=F(x)+C,C=y0−F(x0)y=F(x)+C,\qquad C=y_0-F(x_0)

One derivative defines a family of vertical translations

Conditions. Three antiderivatives of 2x: x² − 2, x² and x² + 2. The arbitrary constant changes only vertical position.

x / dimensionless; y / dimensionless.

All three curves have derivative 2x. A known point chooses one constant: passing through (1,3) requires C = 2 because 1² + C = 3.

Open the interactive lesson to view plotted curves and coordinate tables.

Worked example

A curve has dy/dx=6x2−4x+3dy/dx=6x^2-4x+3 and passes through (1,7)(1,7). Find its equation.

  1. Integrate term by term: y=2x3−2x2+3x+Cy=2x^3-2x^2+3x+C.
  2. Substitute (1,7)(1,7): 7=2−2+3+C7=2-2+3+C.
  3. Hence C=4C=4. Differentiate the resulting equation to recover the stated derivative.

Answer: y=2x3−2x2+3x+4y=2x^3-2x^2+3x+4.

Five common P1 integration mistakes

Use this free ExamLantern worksheet to check indefinite integration with powers and find a curve from a point. Review the five fixes, attempt the questions, then check the worked answers.

Download the free integration worksheet (PDF, 3 pages)

Watch the worked integration lesson on YouTube

For n not equal to -1, the integral of x to the power n is x to the power (n + 1), divided by (n + 1), plus C. Work on an interval where the expression is defined. Check an antiderivative by differentiating it.

Read the five integration fixes

1 Forgetting the constant of integration

Common error

∫ 12x² dx = 4x³

Correction

∫ 12x² dx = 4x³ + C

Different constants give the same derivative. Include + C when giving the general indefinite integral.

2 Dividing by the old power

Common error

∫ 6x⁵ dx = (6/5)x⁶ + C

Correction

∫ 6x⁵ dx = x⁶ + C

Increase the power from 5 to 6, then divide by 6. Differentiating x to the power 6 gives 6x to the power 5.

3 Leaving a constant term unchanged

Common error

∫ (4x + 7) dx = 2x² + 7 + C

Correction

∫ (4x + 7) dx = 2x² + 7x + C

The derivative of 7x is 7. A constant term in the integrand becomes a multiple of x; this is separate from + C.

4 Losing the sign with a negative power

Common error

∫ 5/x² dx = 5/x + C

Correction

∫ 5/x² dx = -5/x + C

Rewrite 5/x² as 5x to the power -2. The new power is -1, so divide by -1. This expression requires x not equal to zero.

5 Using the point in the derivative

Common error

For dy/dx = 8x - 3 through (2, 14), use 14 = 8(2) - 3.

Correction

y = 4x² - 3x + C; 14 = 16 - 6 + C; C = 4.

The point gives x and y, not the gradient. Integrate first, then substitute into y. The curve is y = 4x² - 3x + 4.

Try five integration questions

Show your method. Include the constant of integration where needed. For questions involving fractions or roots, use an interval where the integrand is defined.

Question A

Find ∫ (9x² - 4x + 6) dx.

Question B

Find ∫ 14x⁶ dx.

Question C

Find ∫ [(6x³ - 5x² + 12)/x²] dx, for x ≠ 0.

Question D

Find ∫ (3√x + 2/√x) dx, for x > 0.

Question E

A curve has dy/dx = 10x - 4 and passes through (3, 29). Find its equation.

Before checking your answers

  • Did I increase the power and divide by the new power?
  • Did I integrate constant terms and include + C?
  • Did I rewrite fractions and roots carefully?
  • Did I substitute the point after integrating?
  • Does differentiating my answer recover the integrand or derivative?
Read the worked answers

Compare your method as well as your final expression.

Answer A

  1. Integrate each term separately: 9x³/3 - 4x²/2 + 6x + C.
  2. Simplify: 3x³ - 2x² + 6x + C.
  3. Check: its derivative is 9x² - 4x + 6.

3x³ - 2x² + 6x + C

Answer B

  1. Increase the power to 7 and divide by 7: 14x⁷/7 + C.
  2. Simplify: 2x⁷ + C. Its derivative is 14x⁶.

2x⁷ + C

Answer C

  1. Divide every term by x²: 6x - 5 + 12x⁻².
  2. Integrate: 3x² - 5x - 12x⁻¹ + C.
  3. So the answer is 3x² - 5x - 12/x + C. Differentiate to get 6x - 5 + 12/x².

3x² - 5x - 12/x + C, on an interval not containing 0

Answer D

  1. Rewrite the integrand as 3x1/2 + 2x-1/2.
  2. Integrate: [3/(3/2)]x3/2 + [2/(1/2)]x1/2 + C.
  3. Simplify: 2x3/2 + 4x1/2 + C. Differentiation recovers the original integrand.

2x3/2 + 4x1/2 + C, for x > 0

Answer E

  1. Integrate: y = 5x² - 4x + C.
  2. Substitute the point: 29 = 45 - 12 + C, so C = -4.
  3. Hence y = 5x² - 4x - 4. At x = 3, y = 29; differentiating gives 10x - 4.

y = 5x² - 4x - 4

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