Calculus Examples

$2 x^{2} + 4 x - 6 + 3 \tan (x)$

Step 1

Write $2 x^{2} + 4 x - 6 + 3 \tan (x)$ as a function.

$f (x) = 2 x^{2} + 4 x - 6 + 3 \tan (x)$

Step 2

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 3

Set up the integral to solve.

$F (x) = \int 2 x^{2} + 4 x - 6 + 3 \tan (x) d x$

Step 4

Split the single integral into multiple integrals.

$\int 2 x^{2} d x + \int 4 x d x + \int - 6 d x + \int 3 \tan (x) d x$

Step 5

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int x^{2} d x + \int 4 x d x + \int - 6 d x + \int 3 \tan (x) d x$

Step 6

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$2 (\frac{1}{3} x^{3} + C) + \int 4 x d x + \int - 6 d x + \int 3 \tan (x) d x$

Step 7

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$2 (\frac{1}{3} x^{3} + C) + 4 \int x d x + \int - 6 d x + \int 3 \tan (x) d x$

Step 8

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

$2 (\frac{1}{3} x^{3} + C) + 4 (\frac{1}{2} x^{2} + C) + \int - 6 d x + \int 3 \tan (x) d x$

Step 9

Apply the constant rule.

$2 (\frac{1}{3} x^{3} + C) + 4 (\frac{1}{2} x^{2} + C) - 6 x + C + \int 3 \tan (x) d x$

Step 10

Simplify.

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Step 10.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$2 (\frac{x^{3}}{3} + C) + 4 (\frac{1}{2} x^{2} + C) - 6 x + C + \int 3 \tan (x) d x$

Step 10.2

Combine $\frac{1}{2}$ and $x^{2}$ .

$2 (\frac{x^{3}}{3} + C) + 4 (\frac{x^{2}}{2} + C) - 6 x + C + \int 3 \tan (x) d x$

Step 11

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$2 (\frac{x^{3}}{3} + C) + 4 (\frac{x^{2}}{2} + C) - 6 x + C + 3 \int \tan (x) d x$

Step 12

The integral of $\tan (x)$ with respect to $x$ is $\ln (| \sec (x) |)$ .

$2 (\frac{x^{3}}{3} + C) + 4 (\frac{x^{2}}{2} + C) - 6 x + C + 3 (\ln (| \sec (x) |) + C)$

Step 13

Simplify.

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Step 13.1

Simplify.

$\frac{2 x^{3}}{3} + 2 x^{2} - 6 x + 3 \ln (| \sec (x) |) + C$

Step 13.2

Reorder terms.

$\frac{2}{3} x^{3} + 2 x^{2} - 6 x + 3 \ln (| \sec (x) |) + C$

Step 14

The answer is the antiderivative of the function $f (x) = 2 x^{2} + 4 x - 6 + 3 \tan (x)$ .

$F (x) =$ $\frac{2}{3} x^{3} + 2 x^{2} - 6 x + 3 \ln (| \sec (x) |) + C$