Integrals

The One Thing

Integration reverses differentiation: it finds antiderivatives and, for definite integrals, evaluates accumulated or signed quantities over an interval. Correct solutions depend on recognising the integrand’s form, selecting an appropriate technique, and applying constants, limits, absolute values, and domain conditions correctly.

Definitions and Results

• Antiderivative: A function $F(x)$ is an antiderivative of $f(x)$ if $F^{\prime }(x)=f(x)$.
• Indefinite Integral: $\int f(x)dx=F(x)+C$, representing the family of all antiderivatives, where $C$ is an arbitrary constant.
• Arbitrary Constant: $C$ accounts for the fact that functions differing by a constant have the same derivative.
• Definite Integral: ${\int }_a^{b}f(x)dx$ has fixed limits and gives a number representing signed accumulation between $x=a$ and $x=b$.
• Fundamental Theorem of Calculus: If $F^{\prime }(x)=f(x)$, then

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

• Linearity of Integration:

$$\int [af(x)+bg(x)]dx=a\int f(x)dx+b\int g(x)dx.$$

• Area Under a Curve: If $f(x)\ge 0$ on $[a,b]$, then ${\int }_a^{b}f(x)dx$ gives the area between the curve, the $x$-axis, and $x=a,x=b$.
• Signed Area: Regions below the $x$-axis contribute negatively to a definite integral. Geometric area may therefore require splitting the interval and taking absolute values.
• Improper Integral: An integral with an infinite limit or an unbounded integrand, evaluated through limits when the relevant limit exists.
• Reversal Property:

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

• Zero-Interval Property:

$${\int }_a^{a}f(x)dx=0.$$

• Additivity:

$${\int }_a^{b}f(x)dx+{\int }_b^{c}f(x)dx={\int }_a^{c}f(x)dx.$$

• Order Property: If $f(x)\ge g(x)$ on $[a,b]$, then

$${\int }_a^{b}f(x)dx\ge {\int }_a^{b}g(x)dx.$$

• Symmetry: If $f$ is even,

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

• Fundamental Theorem, accumulation form: For continuous $f$,

$$A(x)={\int }_a^{x}f(t)dt\Rightarrow A^{\prime }(x)=f(x).$$

Standard Integral Forms

• Power rule:

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

• Constant rule:

$$\int kdx=kx+C.$$

• Reciprocal rule:

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

• Exponential forms:

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

• Logarithmic form:

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

• Trigonometric forms:

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

• Inverse-trigonometric forms:

$$\int \frac{dx}{a^2+x^2}=\frac{1}{a}{\tan }^{-1}\left(\frac{x}{a}\right)+C,a>0,$$ $$\int \frac{dx}{\sqrt{a^2-x^2}}={\sin }^{-1}\left(\frac{x}{a}\right)+C,a>0.$$

• Inverse-hyperbolic/logarithmic form:

$$\int \frac{dx}{a^2-x^2}=\frac{1}{2a}\ln \left|\frac{a+x}{a-x}\right|+C.$$

• Logarithmic form:

$$\int \frac{dx}{x^2-a^2}=\frac{1}{2a}\ln \left|\frac{x-a}{x+a}\right|+C.$$

• Quadratic form:

$$\int \frac{dx}{\sqrt{x^2+a^2}}=\ln \left|x+\sqrt{x^2+a^2}\right|+C.$$

Worked Methods

1. Standard integration and linearity

• Rewrite the integrand as a sum of powers, constants, exponentials, logarithmic forms, or trigonometric forms.
• Apply the relevant standard formula term by term.
• Use linearity to retain constant multiples.
• Add $C$ for an indefinite integral.
• Differentiate the result to verify it.

For example, terms involving $x^n$, $e^x$, $\sin x$, $\cos x$, or $\frac{f^{\prime }(x)}{f(x)}$ are matched directly to the standard forms above.

2. Definite integration

• Find an antiderivative $F(x)$ such that $F^{\prime }(x)=f(x)$.
• Write

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

• Evaluate the endpoints:

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

• Do not write a separate constant of integration after evaluating a definite integral.

3. Substitution

• Identify a composite expression and set $u=g(x)$.
• Differentiate to obtain $du=g^{\prime }(x)dx$.
• Rewrite the integrand in terms of $u$ and $du$.
• Integrate using a standard form.
• Replace $u$ by $g(x)$.
• For an indefinite integral, add $C$.
• For a definite integral, change the limits to the corresponding $u$-values, or reverse the substitution before applying the original limits.

4. Integration by parts

Integration by parts follows from the product rule: $$\int udv=uv-\int vdu.$$

• Choose $u$ and $dv$.
• Calculate $du$ and $v$.
• Substitute into

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

• Select $u$ so that $\int vdu$ is simpler than the original integral.
• The LIATE guideline commonly prioritises logarithmic, inverse-trigonometric, algebraic, trigonometric, and exponential functions for $u$.
• Add $C$ for an indefinite integral and verify by differentiation.

5. Partial fractions

• Factor the denominator of the rational function.
• Decompose the rational function into simpler fractions.
• Determine the unknown coefficients.
• Integrate each simpler fraction using standard forms, including logarithmic forms where appropriate.
• Combine the result and add $C$.

This method is especially applicable when the denominator factors.

6. Trigonometric identities

• Identify a product or power of trigonometric functions that is not immediately in standard form.
• Apply an appropriate trigonometric identity.
• Rewrite the integrand into standard trigonometric forms.
• Integrate term by term.
• Add $C$ and verify by differentiation.

7. Area under a curve and signed area

• Identify the interval $[a,b]$.
• Determine whether the function is above or below the $x$-axis.
• Evaluate ${\int }_a^{b}f(x)dx$.
• If the graph crosses the axis, split the interval at the crossing points.
• Take absolute values of the separate signed integrals when geometric area, rather than signed accumulation, is required.

8. Area between two curves

For $y=f(x)$ and $y=g(x)$:

• Identify the points of intersection and the interval $[a,b]$.
• Determine which function is uppermost.
• Integrate

$${\int }_a^b[\text{upper function}-\text{lower function}]dx$$ when the curves do not cross within the interval.

• If the curves cross, split the interval and reassess the upper and lower functions.

9. Improper integrals

• Identify the infinite limit or point where the integrand is unbounded.
• Replace the improper endpoint or singular point with a variable.
• Express the integral as a limit.
• Evaluate the limit.
• The improper integral exists only when the relevant limit exists.

Where It Goes Wrong

• Omitting $C$ from an indefinite integral, even though all antiderivatives differ by an arbitrary constant.
• Writing $\ln f(x)$ instead of $\ln |f(x)|$ in reciprocal and logarithmic forms.
• Using the power rule when $n=-1$, despite the required condition $n\ne -1$; the correct result in that case is $\ln |x|+C$.
• Failing to change the limits during substitution in a definite integral, or failing to reverse the substitution before using the original limits.
• Treating a definite integral as ordinary geometric area when the graph lies below or crosses the $x$-axis.
• Choosing the wrong order in the area-between-curves formula, rather than using upper function minus lower function.
• Retaining a separate constant of integration after evaluating a definite integral.
• Applying integration by parts without choosing $u$ and $dv$ so that $\int vdu$ is simpler.
• Failing to verify an indefinite integral by differentiating the result.

What Gets Asked

• Find an indefinite integral using the power, constant, reciprocal, exponential, logarithmic, trigonometric, inverse-trigonometric, inverse-hyperbolic/logarithmic, or quadratic standard forms.
• Evaluate a definite integral using the Fundamental Theorem of Calculus.
• Use linearity, reversal, zero-interval, additivity, order, or symmetry properties.
• Find an antiderivative and verify it by differentiation.
• Apply substitution, including changed limits for definite integrals.
• Apply integration by parts and select $u$ using the LIATE guideline.
• Decompose a rational function using partial fractions and integrate the resulting terms.
• Simplify a trigonometric integrand using identities before integrating.
• Calculate signed area under a curve and distinguish it from geometric area.
• Find the area between two curves using upper function minus lower function.
• Evaluate an improper integral through a limit and determine whether it exists.
• Explain the relationship between integration and differentiation through the Fundamental Theorem of Calculus.