Integrating powers and polynomials

Reversing the power rule

• Differentiating multiplies by the power and reduces it by one. Integrating does the opposite: raise the power by one and divide by the new power.

$$\int x^n\,dx = \frac{x^{n+1}}{n+1} + c \qquad (n \neq -1)$$

• Each part:

  • $\int$ — the integral sign, meaning "anti-differentiate"
  • $dx$ — states that $x$ is the variable being integrated with respect to
  • $c$ — the constant of integration
  • $n \neq -1$ — because dividing by $n + 1 = 0$ is impossible; that case gives a logarithm instead

• $\displaystyle\int x^4\,dx = \frac{x^5}{5} + c$, and $\displaystyle\int x\,dx = \frac{x^2}{2} + c$.

Why there is a constant

• Differentiating destroys constants: $x^3 + 7$, $x^3 - 2$ and $x^3$ all have derivative $3x^2$.
• So anti-differentiating $3x^2$ cannot recover which constant was there. Every possible constant is a valid answer, and $+c$ says so.
• $\int 3x^2\,dx = x^3 + c$ describes an entire family of curves, all identical in shape and stacked vertically.

• The report's warning: the constant "should not be omitted and, additionally, its value need not necessarily be zero". Assuming $c = 0$ without evidence is as wrong as leaving it out.

Constants and sums

• A constant multiplier comes outside:

$$\int a\,f(x)\,dx = a\int f(x)\,dx$$

• Integrate term by term:

$$\int \left(6x^2 - 4x + 5\right)dx = 2x^3 - 2x^2 + 5x + c$$

• Only one $+c$ is needed for the whole expression, not one per term.

• Integrating a constant gives that constant times $x$:

$$\int 7\,dx = 7x + c$$

(which makes sense: $7 = 7x^0$, so raising the power gives $\frac{7x^1}{1}$).

Negative and fractional powers

• The rule works for these too — convert first, then apply it.

• Worked through:

$$\int \frac{1}{x^3}\,dx = \int x^{-3}\,dx = \frac{x^{-2}}{-2} + c = -\frac{1}{2x^2} + c$$

$$\int \sqrt{x}\,dx = \int x^{1/2}\,dx = \frac{x^{3/2}}{3/2} + c = \frac{2}{3}x^{3/2} + c$$

$$\int \frac{1}{\sqrt{x}}\,dx = \int x^{-1/2}\,dx = \frac{x^{1/2}}{1/2} + c = 2\sqrt{x} + c$$

• Note the fraction arithmetic: dividing by $\frac{3}{2}$ is multiplying by $\frac{2}{3}$. The report lists "did not recognise that $\sqrt{x}$ is $x^{1/2}$" and "did not recognise that $\frac{1}{\sqrt{x}}$ is $x^{-1/2}$" among Not Achieved behaviours.

Preparing an expression first

• The rule works on terms, so expand and split before integrating.

• Expand brackets:

$$\int x(3x - 2)\,dx = \int \left(3x^2 - 2x\right)dx = x^3 - x^2 + c$$

• Split a single-term denominator:

$$\int \frac{x^3 + 2x}{x^2}\,dx = \int \left(x + 2x^{-1}\right)dx$$

(note the second term is the $n = -1$ case and needs the logarithm rule, covered later).

• You cannot integrate a product by integrating each factor. $\int x\sin x\,dx$ is not $\frac{x^2}{2} \times (-\cos x)$. There is no product rule for integration in this standard — expand or use the reverse chain rule instead.

Finding the constant from a given point

• If you are told the curve passes through a particular point, the constant can be found:

  • Integrate to get the general expression with $+c$.
  • Substitute the given $x$ and $y$ values.
  • Solve for $c$.
  • Rewrite the full particular solution.

• Worked through — find $y$ given $\dfrac{dy}{dx} = 4x - 3$ and the curve passes through $(2, 5)$:

  • $$y = 2x^2 - 3x + c$$
  • $5 = 2(4) - 3(2) + c = 8 - 6 + c$, so $c = 3$
  • $$y = 2x^2 - 3x + 3$$

Checking your answer

• Differentiate what you got. If it does not return the original expression, the integration is wrong.
• This takes ten seconds and catches almost every technique error.