Calculus Examples

$4 x (1 + \ln (x))$

Step 1

Write $4 x (1 + \ln (x))$ as a function.

$f (x) = 4 x (1 + \ln (x))$

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.

$F (x) = \int 4 x (1 + \ln (x)) d x$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\int 4 x \cdot 1 + 4 x \ln (x) d x$

Step 4.2

Multiply $4$ by $1$ .

$\int 4 x + 4 x \ln (x) d x$

Step 4.3

Simplify $4 \ln (x)$ by moving $4$ inside the logarithm.

$\int 4 x + x \ln (x^{4}) d x$

Step 5

Rewrite $4 x + x \ln (x^{4})$ as $4 x + x (4 \ln (x))$ .

$\int 4 x + x (4 \ln (x)) d x$

Step 6

Split the single integral into multiple integrals.

$\int 4 x d x + \int x (4 \ln (x)) d x$

Step 7

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

$4 \int x d x + \int x (4 \ln (x)) d x$

Step 8

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

$4 (\frac{1}{2} x^{2} + C) + \int x (4 \ln (x)) d x$

Step 9

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

$4 (\frac{1}{2} x^{2} + C) + 4 \int x (\ln (x)) d x$

Step 10

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = x$ .

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

Step 11

Simplify.

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Step 11.1

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

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

Step 11.2

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

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

Step 11.3

Combine $\ln (x)$ and $\frac{x^{2}}{2}$ .

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

Step 11.4

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

$4 (\frac{x^{2}}{2} + C) + 4 (\frac{\ln (x) x^{2}}{2} - \int \frac{x^{2}}{2} \cdot \frac{1}{x} d x)$

Step 11.5

Multiply $\frac{x^{2}}{2}$ by $\frac{1}{x}$ .

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

Step 11.6

Cancel the common factor of $x^{2}$ and $x$ .

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Step 11.6.1

Factor $x$ out of $x^{2}$ .

$4 (\frac{x^{2}}{2} + C) + 4 (\frac{\ln (x) x^{2}}{2} - \int \frac{x \cdot x}{2 x} d x)$

Step 11.6.2

Cancel the common factors.

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Step 11.6.2.1

Factor $x$ out of $2 x$ .

$4 (\frac{x^{2}}{2} + C) + 4 (\frac{\ln (x) x^{2}}{2} - \int \frac{x \cdot x}{x \cdot 2} d x)$

Step 11.6.2.2

Cancel the common factor.

$4 (\frac{x^{2}}{2} + C) + 4 (\frac{\ln (x) x^{2}}{2} - \int \frac{x \cdot x}{x \cdot 2} d x)$

Step 11.6.2.3

Rewrite the expression.

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

Step 12

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 13

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

$4 (\frac{x^{2}}{2} + C) + 4 (\frac{\ln (x) x^{2}}{2} - \frac{1}{2} (\frac{1}{2} x^{2} + C))$

Step 14

Simplify.

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Step 14.1

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

$4 (\frac{x^{2}}{2} + C) + 4 (\frac{\ln (x) x^{2}}{2} - \frac{1}{2} (\frac{x^{2}}{2} + C))$

Step 14.2

Simplify.

$2 x^{2} + 4 (\frac{\ln (x) x^{2}}{2} - \frac{1}{2} \cdot \frac{x^{2}}{2}) + C$

Step 14.3

Simplify.

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Step 14.3.1

Multiply $\frac{x^{2}}{2}$ by $\frac{1}{2}$ .

$2 x^{2} + 4 (\frac{\ln (x) x^{2}}{2} - \frac{x^{2}}{2 \cdot 2}) + C$

Step 14.3.2

Multiply $2$ by $2$ .

$2 x^{2} + 4 (\frac{\ln (x) x^{2}}{2} - \frac{x^{2}}{4}) + C$

Step 14.4

Simplify.

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Step 14.4.1

Apply the distributive property.

$2 x^{2} + 4 \frac{\ln (x) x^{2}}{2} + 4 (- \frac{x^{2}}{4}) + C$

Step 14.4.2

Cancel the common factor of $2$ .

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Step 14.4.2.1

Factor $2$ out of $4$ .

$2 x^{2} + 2 (2) \frac{\ln (x) x^{2}}{2} + 4 (- \frac{x^{2}}{4}) + C$

Step 14.4.2.2

Cancel the common factor.

$2 x^{2} + 2 \cdot 2 \frac{\ln (x) x^{2}}{2} + 4 (- \frac{x^{2}}{4}) + C$

Step 14.4.2.3

Rewrite the expression.

$2 x^{2} + 2 (\ln (x) x^{2}) + 4 (- \frac{x^{2}}{4}) + C$

Step 14.4.3

Cancel the common factor of $4$ .

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Step 14.4.3.1

Move the leading negative in $- \frac{x^{2}}{4}$ into the numerator.

$2 x^{2} + 2 (\ln (x) x^{2}) + 4 \frac{- x^{2}}{4} + C$

Step 14.4.3.2

Cancel the common factor.

$2 x^{2} + 2 (\ln (x) x^{2}) + 4 \frac{- x^{2}}{4} + C$

Step 14.4.3.3

Rewrite the expression.

$2 x^{2} + 2 (\ln (x) x^{2}) - x^{2} + C$

Step 14.4.4

Reorder factors in $2 \ln (x) x^{2} - x^{2}$ .

$2 x^{2} + 2 x^{2} \ln (x) - x^{2} + C$

Step 14.4.5

Subtract $x^{2}$ from $2 x^{2}$ .

$x^{2} + 2 x^{2} \ln (x) + C$

Step 15

The answer is the antiderivative of the function $f (x) = 4 x (1 + \ln (x))$ .

$F (x) =$ $x^{2} + 2 x^{2} \ln (x) + C$