P1 5.1, P1 5.2 - Integrating powers

Reversing differentiation and the arbitrary constant

If a curve's gradient function is $2x$, must the curve be $y=x^2$? Consider $y=x^2+4$ and $y=x^2-7$: both also differentiate to $2x$. Differentiation has lost the constant height shift.

For $f(x)=2x$, the whole family is

The constant of integration $C$ is arbitrary: it may be any real number. Each choice gives one curve, and differentiating any of them removes $C$.

[DIAGRAM: asset_name: One gradient function and many curves; asset_slug: constant-family; caption: Vertical shifts change the curve's height but leave its tangent gradients unchanged.; description: The three curves y equals x squared minus two, y equals x squared, and y equals x squared plus two are shown with parallel tangents at x equals one. Their marked points have heights negative one, one and three; each tangent gradient is two.]

The curves in the picture have different heights at $x=1$, but their tangents there all have gradient $2$. This is the information a derivative preserves and a vertical shift changes.

The arbitrary constant belongs to the final function, not to the integrand. There is no remaining integral sign or $\mathrm{d}x$ in the result. If several terms are integrated separately, their constants can be combined into one $C$. Also, $\int0\,\mathrm{d}x=C$: a zero gradient allows any constant height.

Choose the family that includes every vertical shift.

Power rule with n not equal to minus one

To reverse differentiation, increase a power by one and divide by that new power.

Since $\mathrm{d}(x^3)/\mathrm{d}x=3x^2$, an antiderivative of $x^2$ is $x^3/3$. Dividing by $3$ cancels the multiplier that differentiation will introduce. More generally, differentiating $x^{n+1}/(n+1)$ gives $x^n$, provided $n+1\ne0$.

The rule includes negative and fractional powers. For example, $\int x^{-1/2}\,\mathrm{d}x=2x^{1/2}+C$ for $x>0$. The excluded power $-1$ would require division by zero; this power rule cannot integrate $1/x$.

Change only the first exponent from $2$ to $-2$. The new exponent is $-1$, so its coefficient is divided by a negative number.

An integrand with exponent $-2$ is allowed; an integrand with exponent $-1$ is the exception. Do not confuse the exponent before integration with the exponent after it.

Change the first coefficient and check its sign carefully.

Rewriting before integration

The power rule works on separate powers, so expand and divide before using it on a product or a quotient.

You already used this strategy for differentiation. For example, $(x+2)^2=x^2+4x+4$, while division by $\sqrt{x}=x^{1/2}$ reduces each of those exponents by $1/2$. Rewriting changes the form, not the function.

Start with the squared bracket on its own.

Change only the denominator: divide the same square by $\sqrt{x}$. This introduces fractional powers, but not a new integration rule.

The opening question has a family of answers: every $y=x^2+C$ has gradient $2x$. The same arbitrary constant is needed after more complicated integrations too.

Use the same expansion-and-division method with a new bracket.

Quick Recap

• Indefinite integration reverses differentiation and gives a family of functions.
• The constant $C$ records the vertical shift that differentiation cannot detect.
• Increase each exponent by one and divide by that new exponent; this rule excludes an integrand power of $-1$.
• Rewrite products and root denominators as sums of powers first, then differentiate your answer to check it.