1 Integration Techniques

Antiderivatives and basic formulas

Integration measures accumulated change and reverses differentiation when an antiderivative exists. It can describe signed area, displacement, and other accumulated quantities.

An Antiderivative of $f$ is a function $F$ satisfying

Because constants disappear under differentiation, all antiderivatives of $f$ have the form $F(x)+C$. This gives the Indefinite integral:

Important basic formulas include

The constant-multiple and sum rules allow an integrand to be separated:

For example,

Differentiating the result is a direct check.

Takeaway: Find an antiderivative, include $C$ for an indefinite integral, and differentiate the result to verify it.

Definite integrals and signed accumulation

A Definite integral has limits of integration and produces a number:

It represents signed accumulation from $x=a$ to $x=b$. Regions where the function is below the $x$-axis contribute negatively, so signed accumulation can differ from total geometric area.

Useful properties are

and

A definite integral does not include $C$, because any constant cancels when the endpoint values are subtracted.

Takeaway: Pay attention to orientation, signs, and whether the problem asks for signed accumulation or geometric area.

The Fundamental Theorem in action

The Fundamental Theorem of Calculus has two complementary parts.

First, an accumulation function has the original integrand as its derivative. If $f$ is continuous and

then

The variable $t$ is a dummy variable that keeps the variable endpoint $x$ separate from the variable of integration. If the upper endpoint is itself a function, the chain rule is also required:

For example,

Second, a definite integral can be evaluated using any antiderivative $F$:

For instance,

Takeaway: Differentiation undoes accumulation, while endpoint subtraction evaluates accumulated change.

Reversing the chain and product rules

Substitution is most effective when an integrand contains a composite expression together with its derivative. Set

Then rewrite the entire integral in terms of $u$, integrate, and substitute back if the integral is indefinite.

For example,

uses $u=3x^2+4$ and $du=6x\,dx$. Therefore,

For a definite integral, either return to $x$ before applying the original limits or transform the limits:

Integration by parts is based on the product rule:

Choose $u$ so that differentiating it simplifies the expression, and choose $dv$ so that it can be integrated readily. A common guideline is LIATE: logarithmic, inverse trigonometric, algebraic, trigonometric, exponential.

For example, with $u=x$ and $dv=\cos x\,dx$, we have $du=dx$ and $v=\sin x$. Thus,

Takeaway: Use substitution for a chain-rule pattern and integration by parts for products or factors that simplify when differentiated.

Trigonometric integrals and identities

Trigonometric identities often turn a difficult trigonometric integral into a substitution or a standard formula.

For

save one factor of $\sin x$ when $m$ is odd and convert the remaining even power using $\sin^2x=1-\cos^2x$. When $n$ is odd, save one factor of $\cos x$ and use $\cos^2x=1-\sin^2x$. If both powers are even, use

For example,

can be rewritten as $\sin x(1-\cos^2x)\cos^2x$. With $u=\cos x$, the result is

For tangent and secant, use

An even power of $\sec x$ suggests reserving $\sec^2x$ for $u=\tan x$, while an odd power of $\tan x$ can suggest reserving $\sec x\tan x$ for $u=\sec x$. Frequently used results include

Takeaway: Inspect parity first, preserve a factor that matches a derivative, and use identities to convert the remaining powers.

Radicals and rational functions

Trigonometric substitution is designed for radicals with one of three quadratic patterns:

• For $\sqrt{a^2-x^2}$, use $x=a\sin\theta$ and $1-\sin^2\theta=\cos^2\theta$.

• For $\sqrt{a^2+x^2}$, use $x=a\tan\theta$ and $1+\tan^2\theta=\sec^2\theta$.

• For $\sqrt{x^2-a^2}$, use $x=a\sec\theta$ and $\sec^2\theta-1=\tan^2\theta$.

After integrating in $\theta$, use a right triangle or the original substitution to return to $x$.

For example, consider

Set $x=3\tan\theta$. Then $dx=3\sec^2\theta\,d\theta$ and $\sqrt{9+x^2}=3\sec\theta$, so

Since $\tan\theta=x/3$ and $\sec\theta=\sqrt{x^2+9}/3$, the result is

Partial fractions applies to rational functions after polynomial division when the numerator degree is not less than the denominator degree. Decompose the rational function according to its factor types, solve for the unknown coefficients, and integrate each simpler term.

For example,

so

Takeaway: Match each radical or denominator structure to its standard decomposition or substitution pattern.

Numerical methods and technique selection

Numerical integration estimates a definite integral from function values. Divide $[a,b]$ into $n$ subintervals using

The midpoint rule uses the center of each subinterval:

The trapezoidal rule connects neighboring points with line segments:

Simpson’s rule uses parabolic approximations over pairs of subintervals and therefore requires $n$ to be even:

For $\int_0^1x^2\,dx$ with the trapezoidal rule and $n=4$, $\Delta x=1/4$, and the estimate is

The exact value is

so the absolute error is approximately $0.01042$. Increasing $n$ generally improves the estimate, and Simpson’s rule is often more accurate for smooth functions.

When selecting a technique, simplify first, then look for a direct formula, substitution, integration by parts, trigonometric identities, trigonometric substitution, partial fractions, or numerical integration. Always check an antiderivative by differentiating it and check definite-integral signs and endpoint values.

Takeaway: Use exact techniques when practical; use numerical rules when an exact antiderivative is unavailable or unnecessary, and compare estimates with exact values when possible.