Calculus Examples

$\frac{d y}{d x} = \frac{3 x^{2} - 4 x}{25}$

Step 1

Rewrite the equation.

$d y = \frac{3 x^{2} - 4 x}{25} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{3 x^{2} - 4 x}{25} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{3 x^{2} - 4 x}{25} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $\frac{1}{25}$ is constant with respect to $x$ , move $\frac{1}{25}$ out of the integral.

$y + C_{1} = \frac{1}{25} \int 3 x^{2} - 4 x d x$

Step 2.3.2

Split the single integral into multiple integrals.

$y + C_{1} = \frac{1}{25} (\int 3 x^{2} d x + \int - 4 x d x)$

Step 2.3.3

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

$y + C_{1} = \frac{1}{25} (3 \int x^{2} d x + \int - 4 x d x)$

Step 2.3.4

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$y + C_{1} = \frac{1}{25} (3 (\frac{1}{3} x^{3} + C_{2}) + \int - 4 x d x)$

Step 2.3.5

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

$y + C_{1} = \frac{1}{25} (3 (\frac{1}{3} x^{3} + C_{2}) - 4 \int x d x)$

Step 2.3.6

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

$y + C_{1} = \frac{1}{25} (3 (\frac{1}{3} x^{3} + C_{2}) - 4 (\frac{1}{2} x^{2} + C_{3}))$

Step 2.3.7

Simplify.

$y + C_{1} = \frac{1}{25} (x^{3} - 2 x^{2}) + C_{4}$

Step 2.4

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

$y = \frac{1}{25} (x^{3} - 2 x^{2}) + K$