Why Every Integral Comes With a Mysterious Constant C

Differentiation finds a rate from a quantity; integration works backwards, finding the quantity from its rate. If you know a car's speed at every moment, integration recovers the distance travelled. It starts with recognising which function has a given derivative.

This lesson covers integration as the inverse of differentiation, standard anti-derivatives, powers of sine and cosine, and the reciprocal and exponential functions.

**A function F is an anti-derivative of f when $F^{\prime }(x)=f(x)$, and because adding a constant does not change a derivative, the indefinite integral is written $\int f(x)dx=F(x)+C$, where C is the constant of integration.

Why the constant appears.** $x^2$, $x^2+5$ and $x^2-3$ all have derivative $2x$, so $\int 2xdx=x^2+C$ describes the whole family.

Properties:

- $\frac{d}{dx}\int f(x)dx=f(x)$
- $\int [f(x)\pm g(x)]dx=\int f(x)dx\pm \int g(x)dx$
- $\int kf(x)dx=k\int f(x)dx$

Worked example. A curve passes through $(1,4)$ and has slope $\frac{dy}{dx}=3x^2+2$ at every point. Then $y=x^3+2x+C$, and $4=1+2+C$ gives $C=1$, so $y=x^3+2x+1$.

Check by differentiating. $\frac{d}{dx}(x^3+2x+1)=3x^2+2$.

An everyday example. Knowing how fast water flows into a tank each minute tells you how much water has been added, but not how much was there at the start — that starting amount plays the role of C.

The substance. Every integral can be checked by differentiating — if the derivative of your answer is not the integrand, the answer is wrong.

**Reversing the standard derivatives gives $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$ for $n\ne -1$, $\int \sin xdx=-\cos x+C$, $\int \cos xdx=\sin x+C$, $\int {\sec }^{2}xdx=\tan x+C$ and $\int {\text{cosec}}^{2}xdx=-\cot x+C$.

More standard results:**

- $\int \sec x\tan xdx=\sec x+C$
- $\int \text{cosec}x\cot xdx=-\text{cosec}x+C$
- $\int \sin axdx=-\frac{\cos ax}{a}+C$

Worked example (polynomial).

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

Worked example (roots and reciprocals).

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

Worked example (trigonometric). $\int (3\cos x-2{\sec }^{2}x)dx=3\sin x-2\tan x+C$, and $\int {\tan }^{2}xdx=\int ({\sec }^{2}x-1)dx=\tan x-x+C$.

An everyday example. **A scooter starting from rest with speed $2t+3$ m/s after t seconds** has covered $s=t^2+3t$ metres, so $4^2+3\times 4=28$ m in the first 4 seconds.

The substance. **The power rule fails for $n=-1$** — it would divide by zero, which is why $\int \frac{1}{x}dx$ needs its own result.

**Powers of sine and cosine are integrated by first lowering the power with identities: ${\sin }^{2}x=\frac{1-\cos 2x}{2}$ and ${\cos }^{2}x=\frac{1+\cos 2x}{2}$ for squares, and ${\sin }^{3}x=\frac{3\sin x-\sin 3x}{4}$ and ${\cos }^{3}x=\frac{3\cos x+\cos 3x}{4}$ for cubes.

Worked example (square).**

$$\int {\sin }^{2}xdx=\int \frac{1-\cos 2x}{2}dx=\frac{x}{2}-\frac{\sin 2x}{4}+C$$

Worked example (cube).

$$\int {\cos }^{3}xdx=\int \frac{3\cos x+\cos 3x}{4}dx=\frac{3\sin x}{4}+\frac{\sin 3x}{12}+C$$

Worked example (fourth power). Squaring the identity for ${\cos }^{2}x$ gives ${\cos }^{4}x=\frac{3}{8}+\frac{\cos 2x}{2}+\frac{\cos 4x}{8}$, so

$$\int {\cos }^{4}xdx=\frac{3x}{8}+\frac{\sin 2x}{4}+\frac{\sin 4x}{32}+C$$

Check. Differentiating $\frac{x}{2}-\frac{\sin 2x}{4}$ gives $\frac{1}{2}-\frac{\cos 2x}{2}={\sin }^{2}x$.

An everyday example. The alternating voltage in a household supply follows a sine wave, and the average power it delivers depends on the integral of ${\sin }^2$ — which is why meters report an effective value rather than the peak.

The substance. **$\int {\sin }^{2}xdx$ is not $\frac{{\sin }^{3}x}{3}$** — the power rule applies to x itself, not to a function of x whose derivative is missing.

**The reciprocal and exponential functions integrate as $\int \frac{1}{x}dx=\log |x|+C$, $\int e^{x}dx=e^x+C$, $\int e^{ax}dx=\frac{e^{ax}}{a}+C$ and $\int a^{x}dx=\frac{a^x}{\log a}+C$.

Why the modulus.** For $x<0$, $\frac{d}{dx}\log (-x)=\frac{-1}{-x}=\frac{1}{x}$, so $\log |x|$ covers negative as well as positive x.

Worked example.

$$\int \left(\frac{3}{x}+4e^{2x}-5^x\right)dx=3\log |x|+2e^{2x}-\frac{5^x}{\log 5}+C$$

Worked example 2. $\int \frac{x^2+3x+2}{x}dx=\int \left(x+3+\frac{2}{x}\right)dx=\frac{x^2}{2}+3x+2\log |x|+C$.

Worked example 3. $\int \frac{1}{x+4}dx=\log |x+4|+C$.

An everyday example. **A bacterial culture in a laboratory growing at a rate proportional to $e^{0.5t}$ has its total growth over time found by integrating the exponential.

The substance. Split a fraction before integrating** — dividing each term by x turns an unfamiliar integrand into standard pieces.