Calculus Examples

$\frac{d^{2} y}{d x^{2}} = \frac{1}{x}$

Step 1

Integrate both sides with respect to $x$ .

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Step 1.1

The first derivative is equal to the integral of the second derivative with respect to $x$ .

$\frac{d y}{d x} = \int \frac{1}{x} d x$

Step 1.2

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

$\frac{d y}{d x} = \ln (| x |) + C_{1}$

Step 2

Rewrite the equation.

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

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

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

Step 3.2

Apply the constant rule.

$y + C_{2} = \int \ln (| x |) + C_{1} d x$

Step 3.3

Integrate the right side.

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Step 3.3.1

Split the single integral into multiple integrals.

$y + C_{2} = \int \ln (| x |) d x + \int C_{1} d x$

Step 3.3.2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (| x |)$ and $d v = 1$ .

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

Step 3.3.3

Simplify.

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Step 3.3.3.1

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

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

Step 3.3.3.2

Cancel the common factor of $x$ .

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Step 3.3.3.2.1

Cancel the common factor.

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

Step 3.3.3.2.2

Rewrite the expression.

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

Step 3.3.4

Apply the constant rule.

$y + C_{2} = \ln (| x |) x - (x + C_{3}) + \int C_{1} d x$

Step 3.3.5

Apply the constant rule.

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

Step 3.3.6

Simplify.

$y + C_{2} = \ln (| x |) x - x + C_{1} x + C_{5}$

Step 3.3.7

Reorder terms.

$y + C_{2} = x \ln (| x |) + C_{1} x - x + C_{5}$

Step 3.4

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

$y = x \ln (| x |) + C x - x + D$