>Integrals>How to Interpret and Calculate the Indefinite Integral

How to Interpret and Calculate the Indefinite Integral

Integrals are mainly split into two categories: Definite and indefinite integrals. The indefinite integral is the same as the anti-derivative.

A challenge in working with integrals is having to find the expression without determining the constant term. The symbol $C$ that you add to the end of integrals represents this unknown constant of integration. To find the constant, you are dependent on having constraints, such that you can find $C$ by substitution and equation solving.

Theory

The Indefinite Integral

\int f (x) d x = F (x) + C, F^{'} (x) = f (x)

Here, $f (x)$ is called the integrand and $C$ is called the constant of integration.

Example 1

Compute the integral $\int \ln x + e^{x} + x^{3} d x$

$\begin{array}{l} \int \ln x + e^{x} + x^{3} d x & = x \ln x - x + e^{x} \\ + \frac{1}{4} x^{4} + C \end{array}$

$\begin{array}{l} \int \ln x + e^{x} + x^{3} d x = x \ln x - x + e^{x} + \frac{1}{4} x^{4} + C \end{array}$

Example 2

Compute the integral

$\int 3 \cos (3 x) - 4 \sin (2 x) d x$

$\int 3 \cos (3 x) - 4 \sin (2 x) d x$

$\begin{array}{l} \int 3 \cos (3 x) - 4 \sin (2 x) d x \\ = 3 \cdot \frac{1}{3} \sin (3 x) + 4 \cdot \frac{1}{2} \cos (2 x) + C \\ = \sin (3 x) + 2 \cos (2 x) + C \end{array}$

$\begin{array}{l} \int 3 \cos (3 x) - 4 \sin (2 x) d x & = 3 \cdot \frac{1}{3} \sin (3 x) + 4 \cdot \frac{1}{2} \cos (2 x) + C \\ = \sin (3 x) + 2 \cos (2 x) + C \end{array}$

Example 3

Compute the integral

$\int \sin (2 x) + 3 \cos (x) - e^{5 x} d x$

$\int \sin (2 x) + 3 \cos (x) - e^{5 x} d x$

$\begin{array}{l} \int \sin (2 x) + 3 \cos (x) - e^{5 x} d x \\ = - \frac{1}{2} \cos (2 x) + 3 \sin (x) - \frac{1}{5} e^{5 x} + C \end{array}$

$\int \sin (2 x) + 3 \cos (x) - e^{5 x} d x = - \frac{1}{2} \cos (2 x) + 3 \sin (x) - \frac{1}{5} e^{5 x} + C$

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