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Primitives
Differentiation assigns to each function a unique derivative by definition. The inverse process asks whether, for a given function $f(x)$, there exists a function $F(x)$ whose derivative is exactly $f(x)$. Such a function is called a primitive of $f$. Formally, $F(x)$ is a primitive of $f(x)$ on the interval $[a, b]$ if $F$ is differentiable throughout $[a, b]$ and:
Not every function admits a primitive on a given interval. A sufficient condition is continuity: every continuous function on a closed interval $[a, b]$ admits a primitive there. The converse does not hold in general. As an example, if $f(x) = 3x^2$, a primitive is $F(x) = x^3$, since:
Unlike derivatives, primitives are not unique. Since the derivative of any constant is zero, the functions $x^3$, $x^3 + 5$, and $x^3 - \frac{1}{2}$ are all primitives of $3x^2$. More generally, if $F(x)$ is a primitive of $f(x)$, then so is $F(x) + c$ for any $c \in \mathbb{R}$, since:
Conversely, any two primitives of the same function differ by a constant. If $F_1(x)$ and $F_2(x)$ are both primitives of $f(x)$, then:
which implies $F_1(x) - F_2(x) = c$ for some constant $c \in \mathbb{R}$.
What is the indefinite integral
The indefinite integral of a function $f(x)$ is the set of all its primitives. Since any two primitives differ by a constant, the entire family is expressed as $F(x) + c$ for $c \in \mathbb{R}$, and is denoted:
From this definition it follows directly that:
Differentiating an indefinite integral returns the original function. This relationship is made precise by the Fundamental Theorem of Calculus, which establishes the formal connection between differentiation and integration.
Example 1
Find the primitive of $f(x) = 3x$ that passes through the point $(2, 1)$. The general primitive is obtained by integrating:
To determine the constant, the condition $F(2) = 1$ is imposed:
The unique primitive satisfying the given a is $F(x) = \frac{3}{2}x^2 - 5$.
Linearity properties
The indefinite integral is a linear operator. The integral of a sum of integrable functions equals the sum of their integrals:
A constant factor can be moved outside the integral sign:
Example 2
Compute the integral of $f(x) = 3x^2 + 2x$. Applying property $1$, the integral splits into two terms, each of which falls under the power rule:
The two integration constants arising from each term combine into a single arbitrary constant, giving $x^3 + x^2 + c$ with $c \in \mathbb{R}$.
Example 3
Compute the integral of $f(x) = 5\sin(x)$. Applying property $2$, the constant factor is moved outside the integral:
Since the integral of $\sin(x)$ is $-\cos(x)$, the result is $-5\cos(x) + c$ with $c \in \mathbb{R}$.
Integral of a power function
The integral of a power function $x^a$ with $a \in \mathbb{R}$ and $a \neq -1$ is given by:
The case $a = -1$ requires a separate treatment and is discussed in the next section.
Example 4
Compute the following integral:
Applying linearity, the constant factors are moved outside and the power rule is applied to each term:
The result is $\dfrac{3}{5}x^5 + \dfrac{5}{3}x^3 + c$ with $c \in \mathbb{R}$.
Example 5
Compute the following integral:
Applying linearity, the integral splits into three terms:
The first term follows directly from the power rule: $\int 4x^3 \, dx = x^4$. For the second, rewriting $\frac{1}{\sqrt{x}}$ as $x^{-1/2}$ and applying the power rule with $a = -\frac{1}{2}$ gives $\int 3x^{-1/2} \, dx = 6\sqrt{x}$. The third term follows from the standard integral of the cosine: $\int 2\cos x \, dx = 2\sin x$.
Assembling the three contributions:
The result can be verified by differentiating $x^4 - 6\sqrt{x} + 2\sin x + c$ term by term, which returns the original integrand.
The logarithmic integral
When $a = -1$, the power rule formula produces a zero denominator and does not apply. The integral in this case is:
This follows from the fact that $\frac{d}{dx} \ln |x| = \frac{1}{x}$, which holds for all $x \neq 0$. The absolute value is necessary because $\ln$ is defined only for positive arguments, while $\frac{1}{x}$ is defined on both $(-\infty, 0)$ and $(0, +\infty)$.
The identity $\int \frac{1}{x} \, dx = \ln |x| + c$ holds separately on $(-\infty, 0)$ and on $(0, +\infty)$. On each interval the arbitrary constant may take a different value, so the most general antiderivative of $\frac{1}{x}$ on its full domain is not a single expression $\ln |x| + c$ with one constant, but a piecewise family with independent constants on the two components.
Fundamental integration rules
| Linearity | \[ \int \left( f(x) + g(x)\right)\, dx = \int f(x)\, dx + \int g(x)\, dx \] |
| Linearity | \[ \int k\, f(x)\, dx = k \int f(x)\, dx \] |
| Power rule | \[ \int x^a\, dx = \dfrac{x^{a+1}}{a+1} + c \quad a \neq -1 \] |
| Logarithmic case | \[ \int \dfrac{1}{x} \,dx = \ln|x| + c \] |
Common Integrals
Below is a summary of the most common basic integrals, useful in calculus and for transforming complex expressions into simpler, well-known forms.
\[\int \frac{1}{x} \, dx = \ln |x| + c\] Further reading
\[\int a^x \, dx = \frac{1}{\ln a} \cdot a^x + c\] Further reading
\[\int \sin x \, dx = -\cos x + c\] Further reading
\[\int \cos x \, dx = \sin x + c\] Further reading
\[\int \frac{1}{\sin^2x}\, dx = \cot x +c\] Further reading
\[\int \frac{1}{\cos^2x}\, dx = \tan x +c\] Further reading
\[\int \sec^2 x \, dx = \tan x + c\] Further reading
\[\int \sec x \tan x \, dx = \sec x + c\] Further reading
\[\int \csc^2 x \, dx = -\cot x + c\] Further reading
\[\int \csc x \cot x \, dx = -\csc x + c\] Further reading
\[\int \frac{dx}{1 + x^2} = \arctan x + c\]
\[\int \frac{dx}{\sqrt{1 - x^2}} = \arcsin x + c\]
These identities hold on any interval where the integrand is defined and continuous.