Integrals

The One Thing

Integration is the reverse process of differentiation: it finds antiderivatives and measures accumulated change. Indefinite integrals produce a family of functions, whereas definite integrals produce a signed numerical value evaluated using the Fundamental Theorem of Calculus.

Definitions and Results

• Antiderivative: A function $F(x)$ is an antiderivative of $f(x)$ if $F^{\prime }(x)=f(x)$.

• Indefinite integral: The collection of all antiderivatives of $f(x)$, written

$$\int f(x)dx=F(x)+C,$$ where $C$ is an arbitrary constant.

• Constant of integration: The constant $C$ is required because functions differing only by a constant have the same derivative.

• Definite integral: The integral

$${\int }_a^{b}f(x)dx$$ gives a signed area or accumulated change over the interval from $a$ to $b$.

• Fundamental Theorem of Calculus: If $F^{\prime }(x)=f(x)$, then

$${\int }_a^{b}f(x)dx=F(b)-F(a).$$ For a non-negative function, this represents the area between the curve, the $x$-axis, $x=a$, and $x=b$.

• Standard integrals: Important results include

$$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C,n\ne -1,$$ $$\int \frac{1}{x}dx=\ln |x|+C,$$ $$\int e^{x}dx=e^x+C,$$ and $$\int a^{x}dx=\frac{a^x}{\ln (a)}+C,a>0,a\ne 1.$$

• Basic trigonometric integrals:

$$\int \sin xdx=-\cos x+C,\int \cos xdx=\sin x+C,$$ $$\int {\sec }^{2}xdx=\tan x+C,\int {\mathrm{cosec}}^{2}xdx=-\cot x+C.$$

• Additional trigonometric forms:

$$\int \sec x\tan xdx=\sec x+C,$$ $$\int \mathrm{cosec}x\cot xdx=-\mathrm{cosec}x+C.$$

• Useful inverse-trigonometric forms:

$$\int \frac{dx}{x^2+a^2}=\frac{1}{a}{\tan }^{-1}\left(\frac{x}{a}\right)+C,$$ $$\int \frac{dx}{\sqrt{a^2-x^2}}={\sin }^{-1}\left(\frac{x}{a}\right)+C.$$

• Linearity:

$$\int [af(x)+bg(x)]dx=a\int f(x)dx+b\int g(x)dx,$$ where $a$ and $b$ are constants.

• Substitution method: If $u=g(x)$, then

$$\int f(g(x))g^{\prime }(x)dx=\int f(u)du.$$

• Integration by parts: Based on the product rule,

$$\int udv=uv-\int vdu.$$ The LIATE guideline can help select $u$: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential.

• Partial fractions: A rational function is expressed as a sum of simpler fractions before integration. The denominator should first be factored, and the degree of the numerator must be less than the degree of the denominator.

• Properties of definite integrals:

$${\int }_a^{a}f(x)dx=0,$$ $${\int }_a^{b}f(x)dx=-{\int }_b^{a}f(x)dx,$$ $${\int }_a^{b}f(x)dx={\int }_a^{c}f(x)dx+{\int }_c^{b}f(x)dx.$$

• Symmetry: If $f(x)$ is even,

$${\int }_{-a}^{a}f(x)dx=2{\int }_0^{a}f(x)dx.$$ If $f(x)$ is odd, $${\int }_{-a}^{a}f(x)dx=0.$$

• Signed area: A definite integral can be negative when the graph lies below the $x$-axis. Geometric area may therefore require splitting the interval and taking absolute values.

Worked Methods

1. Direct integration using standard forms

• Express the integrand as a sum, difference, constant multiple, or standard function.
• Apply linearity.
• Use the appropriate standard integral.
• Add $C$ to an indefinite integral.
• Differentiate the result to check it.

For example, powers are integrated using $$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C,n\ne -1,$$ whereas the reciprocal requires the separate rule $$\int \frac{1}{x}dx=\ln |x|+C.$$

The exponential and trigonometric forms are handled directly: $$\int e^{x}dx=e^x+C,\int a^{x}dx=\frac{a^x}{\ln (a)}+C,$$ with $a>0$ and $a\ne 1$, together with $$\int \sin xdx=-\cos x+C,\int \cos xdx=\sin x+C,$$ $$\int {\sec }^{2}xdx=\tan x+C,\int {\mathrm{cosec}}^{2}xdx=-\cot x+C.$$

2. Substitution

• Identify a composite expression $g(x)$ and set

$$u=g(x).$$

• Differentiate to obtain $du=g^{\prime }(x)dx$.
• Rewrite the integral entirely in terms of $u$ and $du$.
• Integrate using a standard formula.
• Substitute $g(x)$ back for $u$.
• Add $C$ if the integral is indefinite.

The general pattern is $$\int f(g(x))g^{\prime }(x)dx=\int f(u)du.$$

The method is appropriate when part of the integrand is a function whose derivative also appears as a factor.

3. Integration by parts

• Choose $u$ and $dv$.
• Calculate

$$du\text{and}v=\int dv.$$

• Apply

$$\int udv=uv-\int vdu.$$

• Evaluate the remaining integral using standard methods, substitution, or further integration by parts.
• Add $C$ for an indefinite integral.

When choosing $u$, the LIATE guideline is useful in the order: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential.

4. Partial fractions

• Confirm that the integrand is a rational function.
• Check that the degree of the numerator is less than the degree of the denominator.
• Factor the denominator completely.
• Decompose the rational function into simpler fractions.
• Integrate each fraction separately, using logarithmic or other standard forms.
• Combine the result and add $C$ for an indefinite integral.

5. Definite integration using the Fundamental Theorem of Calculus

• Find an antiderivative $F(x)$ such that

$$F^{\prime }(x)=f(x).$$

• Evaluate $F(x)$ at the upper limit $b$ and lower limit $a$.
• Subtract:

$${\int }_a^{b}f(x)dx=F(b)-F(a).$$

• Do not add a constant of integration in the final definite evaluation.

The limits must be applied in the stated order. Reversing them changes the sign: $${\int }_a^{b}f(x)dx=-{\int }_b^{a}f(x)dx.$$

6. Evaluating areas and using symmetry

• Determine whether the function is non-negative over the interval.
• If the graph crosses the $x$-axis, split the interval at the crossing points.
• Evaluate each definite integral.
• Take absolute values where geometric area, rather than signed accumulation, is required.
• Use symmetry when applicable:

- for even $f(x)$, $${\int }_{-a}^{a}f(x)dx=2{\int }_0^{a}f(x)dx;$$ - for odd $f(x)$, $${\int }_{-a}^{a}f(x)dx=0.$$

Where It Goes Wrong

• Omitting $C$ from an indefinite integral, even though additive constants have the same derivative.
• Applying the power rule when $n=-1$; the case $\int 1/xdx$ requires $\ln |x|+C$.
• Failing to include the conditions $a>0$ and $a\ne 1$ for

$$\int a^{x}dx=\frac{a^x}{\ln (a)}+C.$$

• Adding a constant of integration to the final evaluation of a definite integral.
• Reversing or mishandling definite-integral bounds, which changes the sign.
• Treating a signed integral as geometric area when the graph lies below the $x$-axis, or failing to split the interval where the graph changes sign.

What Gets Asked

This material supports questions requiring students to:

• Find indefinite integrals using the power, reciprocal, exponential, trigonometric, and inverse-trigonometric standard forms.
• Apply linearity to integrate sums and constant multiples.
• Use substitution for composite functions.
• Use integration by parts for products, selecting $u$ with the LIATE guideline.
• Decompose suitable rational functions using partial fractions and integrate the resulting terms.
• Evaluate definite integrals using

$${\int }_a^{b}f(x)dx=F(b)-F(a).$$

• Use the properties of definite integrals, including reversed limits, interval splitting, and zero-width intervals.
• Exploit even and odd symmetry on intervals of the form $[-a,a]$.
• Interpret a definite integral as signed area or accumulated change.
• Determine geometric area when a curve lies below the $x$-axis.
• Verify an indefinite integral by differentiating the obtained antiderivative.