Integration — Rules and Examples

Integration means either finding an antiderivative or adding up change. In most first calculus problems, "integrate this function" means: find a function whose derivative is the integrand.

For an indefinite integral,

$$\int f(x)\,dx = F(x) + C$$

means that $F'(x) = f(x)$. The $+C$ matters because any two antiderivatives of the same function can differ by a constant.

If the integral has bounds, such as $\int_a^b f(x)\,dx$, it describes net accumulation on the interval $[a,b]$. In geometry, that often represents signed area. In applications, it can represent a quantity building up over time.

Which Integration Rule Should You Try First?

Start by looking at the shape of the integrand.

• If the integrand is a sum or difference, integrate term by term.
• If there is a constant multiple, pull the constant outside.
• If the expression matches a standard pattern, use the matching antiderivative rule.
• If the integrand is a product, quotient, or composition, a basic rule may not be enough.

This matters because integration is less mechanical than differentiation. There is no single rule that handles every expression directly.

Basic Integration Rules You Should Know

Constant Multiple And Sum Rules

If $a$ and $b$ are constants, then:

$$\int \left(af(x) + bg(x)\right)\,dx = a\int f(x)\,dx + b\int g(x)\,dx$$

This is why term-by-term integration works.

Power Rule

If $n \ne -1$, then:

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

Example: $\int x^4\,dx = \frac{x^5}{5} + C$.

The Special Case $\int \frac{1}{x}\,dx$

When the exponent is $-1$, the power rule does not apply. Instead,

$$\int \frac{1}{x}\,dx = \ln|x| + C$$

The absolute value matters because $\frac{d}{dx}\ln|x| = \frac{1}{x}$ for $x \ne 0$.

Common Standard Antiderivatives

$$\int e^x\,dx = e^x + C$$ $$\int \sin x\,dx = -\cos x + C$$ $$\int \cos x\,dx = \sin x + C$$

These are worth recognizing on sight because they appear often in early integration problems.

Why Integration Feels Like Reversing Differentiation

Differentiation asks, "How is this function changing right now?" Integration asks the reverse question: "What function could have produced this rate of change?"

That is why checking an integral by differentiating your answer is so useful. If the derivative brings you back to the original integrand, the antiderivative is correct.

Integration Example: Combine Three Basic Rules

Find

$$\int \left(4x^3 - \frac{6}{x} + 2\cos x\right)\,dx$$

This is a sum, so integrate each term separately.

For the first term, use the power rule:

$$\int 4x^3\,dx = 4 \cdot \frac{x^4}{4} = x^4$$

For the second term, use the special logarithm case:

$$\int -\frac{6}{x}\,dx = -6\ln|x|$$

For the third term, use the standard trig rule:

$$\int 2\cos x\,dx = 2\sin x$$

Now combine the results:

$$\int \left(4x^3 - \frac{6}{x} + 2\cos x\right)\,dx = x^4 - 6\ln|x| + 2\sin x + C$$

Check by differentiating:

$$\frac{d}{dx}\left(x^4 - 6\ln|x| + 2\sin x + C\right) = 4x^3 - \frac{6}{x} + 2\cos x$$

This check is especially good at catching missed signs and missing constants.

Common Integration Mistakes

1. Forgetting $+C$ in an indefinite integral.
2. Using the power rule for $x^{-1}$. That case is $\ln|x| + C$, not a power-rule answer.
3. Splitting a product as if $\int f(x)g(x)\,dx = \int f(x)\,dx \int g(x)\,dx$. In general, that is false.
4. Copying a derivative fact backward without checking the sign. For example, $\int \sin x\,dx = -\cos x + C$.

When Definite Integrals Are Used

Integration appears whenever a quantity is built from many small changes.

• In geometry, a definite integral can represent signed area under a curve.
• In physics, integrating velocity gives displacement on an interval.
• In economics or engineering, integration can model accumulated cost, growth, or flow.

The condition matters. For example, if velocity changes sign, integrating velocity gives net displacement, not total distance.

When Basic Rules Stop Working

Basic rules work well when the integrand already matches a familiar pattern. If it does not, you may need substitution, integration by parts, or another technique.

That is a useful stopping point: if a formula does not match cleanly, do not force it.

Try A Similar Integral

Try

$$\int \left(5x^2 + \frac{3}{x} - 4\sin x\right)\,dx$$

Then differentiate your answer to check it. If you can explain why the middle term becomes a logarithm, you understand the key exception to the power rule.