Calculus Examples

$\int (\frac{6}{x^{2}} - 5 e^{x} + \sec (x) \tan (x)) d x$

Step 1

Remove parentheses.

$\int \frac{6}{x^{2}} - 5 e^{x} + \sec (x) \tan (x) d x$

Step 2

Split the single integral into multiple integrals.

$\int \frac{6}{x^{2}} d x + \int - 5 e^{x} d x + \int \sec (x) \tan (x) d x$

Step 3

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int \frac{1}{x^{2}} d x + \int - 5 e^{x} d x + \int \sec (x) \tan (x) d x$

Step 4

Apply basic rules of exponents.

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Step 4.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$6 \int {(x^{2})}^{- 1} d x + \int - 5 e^{x} d x + \int \sec (x) \tan (x) d x$

Step 4.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$6 \int x^{2 \cdot - 1} d x + \int - 5 e^{x} d x + \int \sec (x) \tan (x) d x$

Step 4.2.2

Multiply $2$ by $- 1$ .

$6 \int x^{- 2} d x + \int - 5 e^{x} d x + \int \sec (x) \tan (x) d x$

Step 5

By the Power Rule, the integral of $x^{- 2}$ with respect to $x$ is $- x^{- 1}$ .

$6 (- x^{- 1} + C) + \int - 5 e^{x} d x + \int \sec (x) \tan (x) d x$

Step 6

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

$6 (- x^{- 1} + C) - 5 \int e^{x} d x + \int \sec (x) \tan (x) d x$

Step 7

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

$6 (- x^{- 1} + C) - 5 (e^{x} + C) + \int \sec (x) \tan (x) d x$

Step 8

Since the derivative of $\sec (x)$ is $\sec (x) \tan (x)$ , the integral of $\sec (x) \tan (x)$ is $\sec (x)$ .

$6 (- x^{- 1} + C) - 5 (e^{x} + C) + \sec (x) + C$

Step 9

Simplify.

$- \frac{6}{x} - 5 e^{x} + \sec (x) + C$