Calculus Examples

$f (x) = \frac{1}{1 + x^{2}} - e^{x} + \frac{5}{x} + 1$

Step 1

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 2

Set up the integral to solve.

$F (x) = \int \frac{1}{1 + x^{2}} - e^{x} + \frac{5}{x} + 1 d x$

Step 3

Split the single integral into multiple integrals.

$\int \frac{1}{1 + x^{2}} d x + \int - e^{x} d x + \int \frac{5}{x} d x + \int d x$

Step 4

Rewrite $1$ as $1^{2}$ .

$\int \frac{1}{1^{2} + x^{2}} d x + \int - e^{x} d x + \int \frac{5}{x} d x + \int d x$

Step 5

The integral of $\frac{1}{1^{2} + x^{2}}$ with respect to $x$ is $\arctan (x) + C$ .

$\arctan (x) + C + \int - e^{x} d x + \int \frac{5}{x} d x + \int d x$

Step 6

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\arctan (x) + C - \int e^{x} d x + \int \frac{5}{x} d x + \int d x$

Step 7

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$\arctan (x) + C - (e^{x} + C) + \int \frac{5}{x} d x + \int d x$

Step 8

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$\arctan (x) + C - (e^{x} + C) + 5 \int \frac{1}{x} d x + \int d x$

Step 9

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$\arctan (x) + C - (e^{x} + C) + 5 (\ln (| x |) + C) + \int d x$

Step 10

Apply the constant rule.

$\arctan (x) + C - (e^{x} + C) + 5 (\ln (| x |) + C) + x + C$

Step 11

Simplify.

$\arctan (x) - e^{x} + 5 \ln (| x |) + x + C$

Step 12

The answer is the antiderivative of the function $f (x) = \frac{1}{1 + x^{2}} - e^{x} + \frac{5}{x} + 1$ .

$F (x) =$ $\arctan (x) - e^{x} + 5 \ln (| x |) + x + C$