Topic 6.14 Notes – Selecting Techniques for Antidifferentiation

Selecting a Technique for Antidifferentiation

When you see an integral, pause for a few seconds and classify it. Most errors on quizzes happen because students rush into the wrong method.

Ask yourself:

1. Is this already a basic derivative pattern?
2. Is there an inside function with its derivative nearby?
3. Is this a rational function (polynomial over polynomial)?
4. Does the denominator look like $1+x^2$ or $\sqrt{1-x^2}$?
5. Can I simplify this algebraically first?

That short mental checklist saves a lot of time on both MCQs and FRQs.

Core Antiderivative Patterns to Recognize First

Always check these before doing anything fancy.

Power Rule

$$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \ne -1)$$

Examples:

• $\int x^4 dx = \frac{x^5}{5} + C$
• $\int x^{-3} dx = \frac{x^{-2}}{-2} + C$

If $n=-1$, that’s the special case:

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

Students often try to use the power rule on $x^{-1}$. Don’t.

Exponentials and Logs

• $\int e^x dx = e^x + C$
• $\int a^x dx = \frac{a^x}{\ln a} + C$
• If you see $\frac{f'(x)}{f(x)}$, think $$\int \frac{f'(x)}{f(x)} dx = \ln|f(x)| + C$$

Example:

$$\int \frac{3x^2}{x^3+5} dx$$

Derivative of denominator is $3x^2$. That’s a log pattern.

Answer:

$$\ln|x^3+5| + C$$

That recognition shows up constantly on non-calculator sections.

Basic Trig Integrals

Know these cold:

• $\int \sin x dx = -\cos x + C$
• $\int \cos x dx = \sin x + C$
• $\int \sec^2 x dx = \tan x + C$
• $\int \csc^2 x dx = -\cot x + C$
• $\int \sec x \tan x dx = \sec x + C$
• $\int \csc x \cot x dx = -\csc x + C$

If a trig function appears with its derivative, it’s usually immediate.

Inverse Trig Forms

These depend on very specific denominator shapes.

$$\int \frac{1}{1+x^2} dx = \tan^{-1}(x) + C$$

$$\int \frac{1}{\sqrt{1-x^2}} dx = \sin^{-1}(x) + C$$

$$\int \frac{1}{x\sqrt{x^2-1}} dx = \sec^{-1}(|x|) + C$$

If the denominator almost matches but not quite, you probably need algebra first.

U-Substitution

This is reverse chain rule.

Use it when something is clearly “inside” something else.

Example:

$$\int (5x-2)^6 dx$$

Let $u = 5x-2$.
Then $du = 5dx$, so $dx = \frac{1}{5}du$.

The integral becomes:

$$\frac{1}{5} \int u^6 du$$

Now it’s just the power rule.

Clues you need substitution:

• Powers like $(ax+b)^n$
• Exponentials like $e^{g(x)}$
• Fractions like $\frac{g'(x)}{g(x)}$

If leftover $x$’s remain after substitution, you chose the wrong $u$.

Rational Functions

When you see a fraction of polynomials, simplify before integrating.

Long Division

If degree of numerator ≥ degree of denominator, divide first.

Example:

$$\int \frac{x^2+1}{x} dx$$

Rewrite:

$$\int \left(x + \frac{1}{x}\right) dx$$

Now integrate term by term.

Trying substitution first here wastes time.

Completing the Square

If the denominator has a quadratic that doesn’t factor nicely, rewrite it.

Example:

$$\int \frac{1}{x^2+4x+8} dx$$

Complete the square:

$$x^2+4x+8 = (x+2)^2 + 4$$

Now it matches:

$$\frac{1}{(x+2)^2 + 2^2}$$

That’s an arctan pattern.

This move is common in calculator-active FRQs.

Putting It Together

Most integrals fall into one of these categories:

• Polynomial → power rule
• Obvious trig or exponential → basic formula
• Composite structure → substitution
• Rational function → divide or rewrite
• $1+x^2$ or $\sqrt{1-x^2}$ → inverse trig

The skill is not memorizing more formulas. It’s spotting structure fast.