Calculus Examples

$\frac{d y}{d x} = 4 x + \frac{3}{x}$

Step 1

Rewrite the equation.

$d y = (4 x + \frac{3}{x}) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 4 x + \frac{3}{x} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 4 x + \frac{3}{x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 4 x d x + \int \frac{3}{x} d x$

Step 2.3.2

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

$y + C_{1} = 4 \int x d x + \int \frac{3}{x} d x$

Step 2.3.3

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

$y + C_{1} = 4 (\frac{1}{2} x^{2} + C_{2}) + \int \frac{3}{x} d x$

Step 2.3.4

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

$y + C_{1} = 4 (\frac{1}{2} x^{2} + C_{2}) + 3 \int \frac{1}{x} d x$

Step 2.3.5

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

$y + C_{1} = 4 (\frac{1}{2} x^{2} + C_{2}) + 3 (\ln (| x |) + C_{3})$

Step 2.3.6

Simplify.

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Step 2.3.6.1

Simplify.

$y + C_{1} = 4 (\frac{1}{2}) x^{2} + 3 \ln (| x |) + C_{4}$

Step 2.3.6.2

Simplify.

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Step 2.3.6.2.1

Combine $4$ and $\frac{1}{2}$ .

$y + C_{1} = \frac{4}{2} x^{2} + 3 \ln (| x |) + C_{4}$

Step 2.3.6.2.2

Cancel the common factor of $4$ and $2$ .

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Step 2.3.6.2.2.1

Factor $2$ out of $4$ .

$y + C_{1} = \frac{2 \cdot 2}{2} x^{2} + 3 \ln (| x |) + C_{4}$

Step 2.3.6.2.2.2

Cancel the common factors.

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Step 2.3.6.2.2.2.1

Factor $2$ out of $2$ .

$y + C_{1} = \frac{2 \cdot 2}{2 (1)} x^{2} + 3 \ln (| x |) + C_{4}$

Step 2.3.6.2.2.2.2

Cancel the common factor.

$y + C_{1} = \frac{2 \cdot 2}{2 \cdot 1} x^{2} + 3 \ln (| x |) + C_{4}$

Step 2.3.6.2.2.2.3

Rewrite the expression.

$y + C_{1} = \frac{2}{1} x^{2} + 3 \ln (| x |) + C_{4}$

Step 2.3.6.2.2.2.4

Divide $2$ by $1$ .

$y + C_{1} = 2 x^{2} + 3 \ln (| x |) + C_{4}$

Step 2.4

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

$y = 2 x^{2} + 3 \ln (| x |) + C$