Integrating and Differentiating Polynomials

Integrating and Differentiating Polynomials — Moosa Academy

For a power of $x$ , both operations come down to a single short rule each, and the two rules are mirror images. Differentiating pulls the exponent down and lowers it; integrating raises the exponent and divides by the new one.

Theorem The two rules

Differentiate — bring the exponent down as a coefficient, then subtract one from it:

\frac{d}{dx}\left(x^{n}\right) = n\,x^{\,n-1}

Integrate — add one to the exponent, divide by the new exponent, and add the constant:

\int x^{n} \, dx = \frac{x^{\,n+1}}{n+1} + C

Notice the symmetry: one lowers the degree by one, the other raises it by one. That is the first sign that these operations undo each other.

Example Differentiating: three cases

\dfrac{d}{dx}\left(x^{2}\right)

: bring down 2, exponent becomes 1 ⟹

2x

\dfrac{d}{dx}\left(x^{3}\right)

: bring down 3, exponent becomes 2 ⟹

3x^{2}

\dfrac{d}{dx}\left(x^{6}\right)

: bring down 6, exponent becomes 5 ⟹

6x^{5}

A constant such as 7 can be written $7x^{0}$ . Bringing down the exponent multiplies everything by zero, so the derivative is 0 — which matches the fact that a constant never changes.

Example Integrating: four cases

\displaystyle\int x \, dx

: exponent 1 becomes 2, divide by 2 ⟹

\dfrac{x^{2}}{2} + C

\displaystyle\int x^{3} \, dx

: exponent 3 becomes 4, divide by 4 ⟹

\dfrac{x^{4}}{4} + C

\displaystyle\int x^{5} \, dx

: exponent 5 becomes 6, divide by 6 ⟹

\dfrac{x^{6}}{6} + C

\displaystyle\int 7 \, dx

: write 7 as

7x^{0}

, exponent becomes 1 ⟹

7x + C

The constant case is not an exception. Because $x^{0} = 1$ , the ordinary rule handles it — which is why integrating any constant simply attaches an $x$ .

Example A fractional exponent

Integrate $\sqrt{x}$ . The rule works unchanged once the root is written as a power.

Step 1 — rewrite:

\sqrt{x} = x^{1/2}

Step 2 — raise the exponent:

\tfrac{1}{2} + 1 = \tfrac{3}{2}

Step 3 — divide by the new exponent, that is multiply by

\tfrac{2}{3}

\int \sqrt{x} \, dx = \frac{2}{3}x^{3/2} + C

Dividing by a fraction is the same as multiplying by its reciprocal, so dividing by $\tfrac{3}{2}$ gives the factor $\tfrac{2}{3}$ .

Note The two rules undo each other

Start with $x^{2}$ and apply both operations in turn:

Differentiate:

x^{2} \longrightarrow 2x

Integrate that back:

\displaystyle\int 2x \, dx = x^{2} + C

⟹ back where we started, apart from the constant

Integration raises the degree by one and differentiation lowers it by one, so performing both returns the original function. The only trace left behind is $C$ — the constant that differentiation had erased.

Summary

Differentiate xⁿ: bring n down as a coefficient and lower the exponent by one.
Integrate xⁿ: raise the exponent by one, divide by the new exponent, add C.
d/dx(x³) = 3x², and ∫x³ dx = x⁴/4 + C.
A constant is x⁰: its derivative is 0, and ∫7 dx = 7x + C.
Fractional exponents follow the same rule: ∫√x dx = (2/3)x^(3/2) + C.
The two rules move the degree in opposite directions, so each undoes the other.