Calculus Examples

$\frac{d y}{d x} = 4 x \ln (x)$

Step 1

Rewrite the equation.

$d y = 4 x \ln (x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 4 x \ln (x) d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 4 x \ln (x) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

$y + C_{1} = 4 \int x \ln (x) d x$

Step 2.3.2

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

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

Step 2.3.3

Simplify.

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Step 2.3.3.1

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

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

Step 2.3.3.2

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

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

Step 2.3.3.3

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

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

Step 2.3.3.4

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

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

Step 2.3.3.5

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

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Step 2.3.3.5.1

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

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

Step 2.3.3.5.2

Cancel the common factors.

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Step 2.3.3.5.2.1

Factor $x$ out of $2 x$ .

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

Step 2.3.3.5.2.2

Cancel the common factor.

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

Step 2.3.3.5.2.3

Rewrite the expression.

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

Step 2.3.4

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

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

Step 2.3.5

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

$y + C_{1} = 4 (\frac{\ln (x) x^{2}}{2} - \frac{1}{2} (\frac{1}{2} x^{2} + C_{2}))$

Step 2.3.6

Simplify the answer.

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Step 2.3.6.1

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

$y + C_{1} = 4 (\frac{\ln (x) x^{2}}{2} - \frac{1}{2} (\frac{x^{2}}{2} + C_{2}))$

Step 2.3.6.2

Rewrite $4 (\frac{\ln (x) x^{2}}{2} - \frac{1}{2} (\frac{x^{2}}{2} + C_{2}))$ as $4 (\frac{\ln (x) x^{2}}{2} - \frac{x^{2}}{4}) + C_{2}$ .

$y + C_{1} = 4 (\frac{\ln (x) x^{2}}{2} - \frac{x^{2}}{4}) + C_{2}$

Step 2.3.6.3

Reorder terms.

$y + C_{1} = 4 (\frac{1}{2} \ln (x) x^{2} - \frac{1}{4} x^{2}) + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$y = 4 (\frac{1}{2} \ln (x) x^{2} - \frac{1}{4} x^{2}) + K$