Integral Symbol
\[ \int \]Indicates integration.
Calculus Guide
Integrals Explained
Learn how integrals represent accumulation, reverse differentiation, and measure quantities such as signed area over an interval.
Accumulation and Antiderivatives
What Does an Integral Mean?
Integration is one of the two central operations of calculus.
While derivatives measure instantaneous change, integrals measure accumulation.
Integrals also reverse the process of differentiation by finding antiderivatives.
Integral
\[
\int f(x)\,dx
\]
Integration asks for a function whose derivative is \(f(x)\), or measures accumulated change over an interval.
Big Idea
Integration Reverses Differentiation
If
\[ F'(x)=f(x), \]then
\[ \int f(x)\,dx = F(x)+C. \]The function \(F\) is an antiderivative of \(f\).
Reverse Differentiation
Antiderivatives
An antiderivative of \(f(x)\) is a function whose derivative equals \(f(x)\).
Definition
\[ \boxed{ F'(x)=f(x) } \]means \(F(x)\) is an antiderivative of \(f(x)\).
Example
Find an Antiderivative of \(6x\)
We need a function whose derivative is \(6x\).
Since
\[ \frac{d}{dx}(3x^2)=6x, \]one antiderivative is
\[ 3x^2. \]But so are
\[ 3x^2+1, \qquad 3x^2-5, \qquad 3x^2+100. \]All of these differ only by a constant.
Families of Antiderivatives
Indefinite Integrals
\[
\boxed{
\int f(x)\,dx
=
F(x)+C
}
\]
Integrand
\[ f(x) \]The expression being integrated.
Differential
\[ dx \]Indicates the variable of integration.
Constant of Integration
\[ C \]Represents all possible constant shifts of the antiderivative.
Do Not Forget \(+C\)
Why Do We Add the Constant of Integration?
Function
\[ F(x)=x^2 \] \[ F'(x)=2x \]Shifted Function
\[ G(x)=x^2+7 \] \[ G'(x)=2x \]Why \(+C\)?
Derivatives Lose Constant Information
Every function of the form
\[ x^2+C \]has derivative
\[ 2x. \]Therefore,
\[ \int 2x\,dx = x^2+C. \]Core Rules
Basic Integration Rules
Constant Rule
\[ \int c\,dx = cx+C \]Power Rule
\[ \int x^n\,dx = \frac{x^{n+1}}{n+1}+C \]for \(n\neq -1\)
Constant Multiple
\[ \int c f(x)\,dx = c \int f(x)\,dx \]Sum Rule
\[ \int \left[ f(x)+g(x) \right] dx = \int f(x)\,dx + \int g(x)\,dx \]Difference Rule
\[ \int \left[ f(x)-g(x) \right] dx = \int f(x)\,dx - \int g(x)\,dx \]The Main Algebraic Rule
The Integration Power Rule
\[
\boxed{
\int x^n\,dx
=
\frac{x^{n+1}}{n+1}
+
C
}
\]
provided \(n\neq -1\).
1
Add 1 to the exponent
2
Divide by the new exponent
3
Add \(+C\)
Example
\[ \int x^4\,dx \] \[ = \frac{x^5}{5}+C \]Example
\[ \int x^{-3}\,dx \] \[ = \frac{x^{-2}}{-2}+C \] \[ = -\frac{1}{2x^2}+C \]Example
\[ \int \sqrt{x}\,dx \]Rewrite:
\[ \sqrt{x} = x^{1/2} \] \[ = \frac{x^{3/2}}{3/2}+C \] \[ = \frac{2}{3}x^{3/2}+C \]Integrate Term by Term
Integrating Polynomials
Example
Evaluate
\[ \int \left( 4x^3-6x^2+5x-8 \right) dx. \]Integrate each term:
\[ \int 4x^3\,dx = x^4 \] \[ \int -6x^2\,dx = -2x^3 \] \[ \int 5x\,dx = \frac{5}{2}x^2 \] \[ \int -8\,dx = -8x. \]
\[
\boxed{
x^4
-
2x^3
+
\frac52x^2
-
8x
+
C
}
\]
The Exception to the Power Rule
What About \(\frac{1}{x}\)?
Power Rule Exception
Do Not Use the Power Rule When \(n=-1\)
Since \(x^{-1}=\frac1x\), using the power rule would require dividing by zero.
\[
\boxed{
\int \frac{1}{x}\,dx
=
\ln|x|+C
}
\]
Accumulation Over an Interval
Definite Integrals
A definite integral gives a number rather than a family of functions.
\[
\boxed{
\int_a^b f(x)\,dx
}
\]
Lower Limit
\[ a \]Starting \(x\)-value.
Upper Limit
\[ b \]Ending \(x\)-value.
Integrand
\[ f(x) \]Quantity being accumulated.
Geometric Meaning
Definite Integrals and Signed Area
Above the \(x\)-Axis
\[ f(x)>0 \]Contributes positive area.
Below the \(x\)-Axis
\[ f(x)<0 \]Contributes negative signed area.
Important Distinction
A definite integral gives net signed area. Total geometric area may require splitting the interval where the function crosses the \(x\)-axis.
The Central Connection
Fundamental Theorem of Calculus
The Fundamental Theorem connects differentiation and integration.
Evaluating a Definite Integral
\[ \boxed{ \int_a^b f(x)\,dx = F(b)-F(a) } \]where \(F'(x)=f(x)\).
1
Find an antiderivative \(F(x)\)
2
Evaluate \(F(b)\)
3
Subtract \(F(a)\)
Example
Evaluate
\[ \int_1^3 2x\,dx. \]An antiderivative of \(2x\) is
\[ F(x)=x^2. \]Apply the Fundamental Theorem:
\[ \left[ x^2 \right]_1^3 = 3^2-1^2. \] \[ = 9-1. \]
\[
\boxed{8}
\]
Accumulation Functions
Fundamental Theorem: Differentiating an Integral
\[
\boxed{
\frac{d}{dx}
\left[
\int_a^x f(t)\,dt
\right]
=
f(x)
}
\]
Example
Differentiate
\[ G(x) = \int_2^x \left( t^3+1 \right) dt. \]By the Fundamental Theorem,
\[
\boxed{
G'(x)=x^3+1
}
\]
Dummy Variable
The \(t\) inside the integral is a temporary variable. The result is expressed in terms of the upper limit \(x\).
Variable Upper Bounds
Fundamental Theorem with the Chain Rule
\[
\boxed{
\frac{d}{dx}
\left[
\int_a^{g(x)}
f(t)\,dt
\right]
=
f(g(x))g'(x)
}
\]
Example
\[ G(x) = \int_0^{x^2} \cos t\,dt. \]Evaluate the integrand at the upper bound:
\[ \cos(x^2). \]Then multiply by the derivative of the upper bound:
\[ \frac{d}{dx}(x^2)=2x. \]
\[
\boxed{
G'(x)
=
2x\cos(x^2)
}
\]
Essential Formulas
Basic Trigonometric Integrals
Cosine
\[ \boxed{ \int \cos x\,dx = \sin x+C } \]Sine
\[ \boxed{ \int \sin x\,dx = -\cos x+C } \]Secant Squared
\[ \boxed{ \int \sec^2x\,dx = \tan x+C } \]Cosecant Squared
\[ \boxed{ \int \csc^2x\,dx = -\cot x+C } \]Secant-Tangent
\[ \boxed{ \int \sec x\tan x\,dx = \sec x+C } \]Cosecant-Cotangent
\[ \boxed{ \int \csc x\cot x\,dx = -\csc x+C } \]Reverse Chain Rule
\(u\)-Substitution
\(u\)-substitution simplifies an integral when one part of the integrand is the derivative of another part.
Main Idea
Look for an Inside Function and Its Derivative
If
\[ u=g(x), \]then
\[ du=g'(x)\,dx. \]
1
Choose
Let \(u\) Equal the Inner Expression
2
Differentiate
Find \(du\)
3
Rewrite
Express the Integral in Terms of \(u\)
4
Integrate
Use the Simpler \(u\)-Integral
5
Substitute Back
Replace \(u\) with the Original Expression
Worked Example
\(u\)-Substitution Example
Example
Evaluate
\[ \int 2x(x^2+1)^4\,dx. \]Choose
\[ u=x^2+1. \]Differentiate:
\[ du=2x\,dx. \]Rewrite the integral:
\[ \int u^4\,du. \]Integrate:
\[ \frac{u^5}{5}+C. \]Substitute back:
\[
\boxed{
\frac{(x^2+1)^5}{5}+C
}
\]
Missing a Constant?
Adjusting for Constant Factors
Example
\[ \int x(x^2+4)^3\,dx. \]Let
\[ u=x^2+4. \]Then
\[ du=2x\,dx. \]Therefore,
\[ x\,dx = \frac12\,du. \]Rewrite:
\[ \frac12 \int u^3\,du. \] \[ = \frac12 \cdot \frac{u^4}{4} + C. \] \[ = \frac{u^4}{8}+C. \]
\[
\boxed{
\frac{(x^2+4)^4}{8}+C
}
\]
Substitution with Bounds
\(u\)-Substitution in Definite Integrals
Method 1
Substitute Back to \(x\)
Integrate in \(u\), replace \(u\) with the original expression, then use the original \(x\)-bounds.
Method 2
Change the Bounds
Convert the original \(x\)-bounds into \(u\)-bounds and finish entirely in \(u\).
Example
\[ \int_0^1 2x(x^2+1)^2\,dx. \]Let
\[ u=x^2+1, \qquad du=2x\,dx. \]Change the bounds:
\[ x=0 \Rightarrow u=1 \] \[ x=1 \Rightarrow u=2. \]The integral becomes
\[ \int_1^2u^2\,du. \] \[ = \left[ \frac{u^3}{3} \right]_1^2. \] \[ = \frac83-\frac13. \]
\[
\boxed{
\frac73
}
\]
Useful Properties
Properties of Definite Integrals
Same Bounds
\[ \int_a^a f(x)\,dx = 0 \]Reverse Bounds
\[ \int_b^a f(x)\,dx = - \int_a^b f(x)\,dx \]Split an Interval
\[ \int_a^c f(x)\,dx + \int_c^b f(x)\,dx = \int_a^b f(x)\,dx \]Constant Multiple
\[ \int_a^b cf(x)\,dx = c \int_a^b f(x)\,dx \]Even and Odd Functions
Symmetry in Definite Integrals
Even Function
\[ f(-x)=f(x) \] \[ \boxed{ \int_{-a}^{a} f(x)\,dx = 2 \int_0^a f(x)\,dx } \]Odd Function
\[ f(-x)=-f(x) \] \[ \boxed{ \int_{-a}^{a} f(x)\,dx = 0 } \]A Powerful Check
Check an Indefinite Integral by Differentiating
Suppose you found
\[ \int \left( 3x^2+4 \right) dx = x^3+4x+C. \]Differentiate your answer:
\[ \frac{d}{dx} \left( x^3+4x+C \right) = 3x^2+4. \]
The derivative matches the
original integrand.
Watch Out
Common Integration Mistakes
1
Forgetting \(+C\)
Indefinite integrals represent a family of antiderivatives.
2
Using the Derivative Power Rule
Integration adds one to the exponent and divides by the new exponent.
3
Using the Power Rule on \(\frac1x\)
Remember: \(\int \frac1x\,dx=\ln|x|+C\).
4
Adding \(+C\) to a Definite Integral
A definite integral evaluates to a number, so the constants cancel.
5
Computing \(F(a)-F(b)\)
The Fundamental Theorem uses upper minus lower: \(F(b)-F(a)\).
6
Confusing Net Area with Total Area
Area below the axis contributes negatively to a definite integral.
7
Choosing \(u\) but Not Rewriting Everything
After substitution, the entire integral should be expressed in the new variable.
8
Mixing \(x\)-Bounds and \(u\)-Expressions
Either substitute back to \(x\), or convert the bounds to \(u\).
Problem-Solving Roadmap
Quick Integration Strategy
1
Identify
Indefinite or Definite?
Indefinite integrals need \(+C\). Definite integrals use bounds.
2
Simplify
Rewrite Powers When Helpful
Convert radicals and denominator powers before integrating.
3
Choose a Rule
Basic Formula or Substitution?
Look first for a direct antiderivative rule, then consider \(u\)-substitution.
4
Integrate
Find the Antiderivative
5
Finish
Add \(+C\) or Evaluate the Bounds
6
Check
Differentiate When Possible
For indefinite integrals, differentiating your answer should reproduce the integrand.
Keep This Handy
Integrals Quick Reference
Power Rule
\[ \int x^n\,dx = \frac{x^{n+1}}{n+1} + C \]\(n\neq -1\)
Reciprocal
\[ \int \frac1x\,dx = \ln|x|+C \]Cosine
\[ \int \cos x\,dx = \sin x+C \]Sine
\[ \int \sin x\,dx = -\cos x+C \]Secant Squared
\[ \int \sec^2x\,dx = \tan x+C \]Fundamental Theorem
\[ \int_a^b f(x)\,dx = F(b)-F(a) \]FTC Derivative Form
\[ \frac{d}{dx} \left[ \int_a^x f(t)\,dt \right] = f(x) \]\(u\)-Substitution
\[ u=g(x), \qquad du=g'(x)\,dx \]Reverse Bounds
\[ \int_b^a f(x)\,dx = - \int_a^b f(x)\,dx \]Split Interval
\[ \int_a^c f(x)\,dx + \int_c^b f(x)\,dx = \int_a^b f(x)\,dx \]Before You Practice
Integrals Checklist
- Understand integration as both accumulation and reverse differentiation.
- Recognize an antiderivative as a function whose derivative is the integrand.
- Know the notation for an indefinite integral.
- Include \(+C\) with indefinite integrals.
- Apply the integration power rule correctly.
- Remember the special rule \(\int \frac1x\,dx=\ln|x|+C\).
- Integrate polynomials term by term.
- Understand the notation and meaning of a definite integral.
- Distinguish signed area from total geometric area.
- Use \(F(b)-F(a)\) to evaluate definite integrals.
- Differentiate accumulation functions using the Fundamental Theorem.
- Know the basic trigonometric antiderivatives.
- Recognize when \(u\)-substitution reverses the chain rule.
- Handle constant factors correctly during substitution.
- Change the bounds or substitute back when using \(u\)-substitution in a definite integral.
- Use basic properties of definite integrals, including reversed and split intervals.
- Use symmetry when integrating even or odd functions over symmetric intervals.
- Check indefinite integrals by differentiating your result.
Your Turn
Ready to Practice Integrals?
Practice antiderivatives, indefinite and definite integrals, the Fundamental Theorem of Calculus, \(u\)-substitution, and basic trigonometric integrals.
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