(cod: P-69-44-5) The Fundamental Theorem of Calculus, in its two parts, broadly states that differentiation and integration are inverse processes. In particular, the concept of primitive or antiderivative is central to the study of Differential and Integral Calculus. We say that a function \(F(x)\) is a primitive of \(f(x)\) on an interval \(I\) if \(F'(x) = f(x)\) for all \(x \in I\). Motivated by the Fundamental Theorem of Calculus, we introduce the concept of the indefinite integral of a function \(f\), which represents the general form of a primitive of \(f\) on an interval.

Observe that if two functions \(F_1\) and \(F_2\) are primitives of the same function \(f\) on an open interval \(I\), then \(F_1\) and \(F_2\) have identical derivatives on \(I\) and therefore differ by a constant. Thus, if \(F(x)\) is a primitive of \(f(x)\) on an open interval \(I\), the indefinite integral of \(f\) is defined by \[\int f(x) \, dx = F(x) + C,\] which tells us that the general form of a primitive of \(f\) on an interval \(I\) is given by the function \(F\) plus a constant. Note that the symbol for the indefinite integral is the same as that introduced for the definite integral, except that it does not include the integration limits. Observe that the definite integral of a function on an interval \([a,b]\) is a number, while the indefinite integral represents a family of functions, or the general form of a primitive of \(f\) on an interval.

For example, we know that \(\dfrac{d}{dx}\sin(x) = \cos(x)\). We then have that \[\int \cos(x) \, dx = \sin(x) + C.\] Consider the following statements.

(1) If \(p \ne -1\), then \[\displaystyle \int x^p \, dx = \dfrac{x^{p+1}}{p+1} + C.\]

(2) \begin{eqnarray*} & & \int (x^3 - 2x^2 + 4x -2) \, dx \\ & & \\ &=& \dfrac{x^4}{4} - \dfrac{2x^3}{3} + 2x^2 - 2x + C. \end{eqnarray*}



(3) If \[\small{\begin{eqnarray*}p(x) &=& c_n x^n + c_{n-1}x^{n-1}+ \cdots + c_1x + c_0 \\ & & \\ &=& \sum_{k = 0}^n c_k x^k \end{eqnarray*}}\] is a polynomial function, then \begin{eqnarray*} & & \int p(x) \, dx \\ & & \\ &=& \small{\dfrac{c_n x^{n+1}}{n+1} + \cdots + \dfrac{c_1 x^2}{2} + c_0x + C} \\ & & \\ &=& C + \sum_{k=0}^n \dfrac{c_k x^{k+1}}{k+1}. \end{eqnarray*}

(4) \(\displaystyle \int \dfrac{1}{x} \, dx = \ln|x| + C.\)

(5) \(\displaystyle \int \sin(x) \, dx = - \cos(x) + C\).

(6) \(\displaystyle \int \cos(x) \, dx = \sin(x) + C\).

(7) \(\displaystyle \int \sec^2(x) \, dx = \tan(x) + C\).

(8) If \(0 < b \ne 1\), \(\displaystyle \int b^x \, dx = \dfrac{b^x}{\ln{b}} + C\).<%9%12%>
(9) \(\displaystyle \int e^x \, dx = e^x + C\).

(10) \(\displaystyle \int \text{sinh}\, x \, dx = \cosh{x} + C\).

(11) \(\displaystyle \int \cosh{x} \, dx = \text{sinh} \, {x} + C\).

(12) \(\displaystyle \int \dfrac{1}{1+x^2} \, dx = \arctan{x} + C\).

(13) \(\displaystyle \int \dfrac{1}{\sqrt{1 + t^2}} \, dt = \text{arsinh}\, {t} + C\).

Regarding these statements, we have that: