Calculus Examples

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

Step 1

Integrate both sides with respect to $x$ .

Tap for more steps...

Step 1.1

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

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

Step 1.2

Apply basic rules of exponents.

Tap for more steps...

Step 1.2.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 1.2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

Tap for more steps...

Step 1.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.2.2.2

Multiply $2$ by $- 1$ .

$\frac{d y}{d x} = \int x^{- 2} d x$

Step 1.3

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

$\frac{d y}{d x} = - x^{- 1} + C_{1}$

Step 1.4

Rewrite $- x^{- 1} + C_{1}$ as $- \frac{1}{x} + C_{1}$ .

$\frac{d y}{d x} = - \frac{1}{x} + C_{1}$

Step 2

Rewrite the equation.

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

Step 3

Integrate both sides.

Tap for more steps...

Step 3.1

Set up an integral on each side.

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

Step 3.2

Apply the constant rule.

$y + C_{2} = \int - \frac{1}{x} + C_{1} d x$

Step 3.3

Integrate the right side.

Tap for more steps...

Step 3.3.1

Split the single integral into multiple integrals.

$y + C_{2} = \int - \frac{1}{x} d x + \int C_{1} d x$

Step 3.3.2

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$y + C_{2} = - \int \frac{1}{x} d x + \int C_{1} d x$

Step 3.3.3

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

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

Step 3.3.4

Apply the constant rule.

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

Step 3.3.5

Simplify.

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

Step 3.3.6

Reorder terms.

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

Step 3.4

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

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