Calculus Examples

$\frac{d y}{d x} = {(15 x^{2} - 6)}^{2}$

Step 1

Rewrite the equation.

$d y = {(15 x^{2} - 6)}^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int {(15 x^{2} - 6)}^{2} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int {(15 x^{2} - 6)}^{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Expand ${(15 x^{2} - 6)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(15 x^{2} - 6)}^{2}$ as $(15 x^{2} - 6) (15 x^{2} - 6)$ .

$y + C_{1} = \int (15 x^{2} - 6) (15 x^{2} - 6) d x$

Step 2.3.1.2

Apply the distributive property.

$y + C_{1} = \int 15 x^{2} (15 x^{2} - 6) - 6 (15 x^{2} - 6) d x$

Step 2.3.1.3

Apply the distributive property.

$y + C_{1} = \int 15 x^{2} (15 x^{2}) + 15 x^{2} \cdot - 6 - 6 (15 x^{2} - 6) d x$

Step 2.3.1.4

Apply the distributive property.

$y + C_{1} = \int 15 x^{2} (15 x^{2}) + 15 x^{2} \cdot - 6 - 6 (15 x^{2}) - 6 \cdot - 6 d x$

Step 2.3.1.5

Move $x^{2}$ .

$y + C_{1} = \int 15 \cdot 15 x^{2} x^{2} + 15 x^{2} \cdot - 6 - 6 (15 x^{2}) - 6 \cdot - 6 d x$

Step 2.3.1.6

Move $x^{2}$ .

$y + C_{1} = \int 15 \cdot 15 x^{2} x^{2} + 15 \cdot - 6 x^{2} - 6 (15 x^{2}) - 6 \cdot - 6 d x$

Step 2.3.1.7

Multiply $15$ by $15$ .

$y + C_{1} = \int 225 x^{2} x^{2} + 15 \cdot - 6 x^{2} - 6 \cdot 15 x^{2} - 6 \cdot - 6 d x$

Step 2.3.1.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y + C_{1} = \int 225 x^{2 + 2} + 15 \cdot - 6 x^{2} - 6 \cdot 15 x^{2} - 6 \cdot - 6 d x$

Step 2.3.1.9

Add $2$ and $2$ .

$y + C_{1} = \int 225 x^{4} + 15 \cdot - 6 x^{2} - 6 \cdot 15 x^{2} - 6 \cdot - 6 d x$

Step 2.3.1.10

Multiply $15$ by $- 6$ .

$y + C_{1} = \int 225 x^{4} - 90 x^{2} - 6 \cdot 15 x^{2} - 6 \cdot - 6 d x$

Step 2.3.1.11

Multiply $- 6$ by $15$ .

$y + C_{1} = \int 225 x^{4} - 90 x^{2} - 90 x^{2} - 6 \cdot - 6 d x$

Step 2.3.1.12

Multiply $- 6$ by $- 6$ .

$y + C_{1} = \int 225 x^{4} - 90 x^{2} - 90 x^{2} + 36 d x$

Step 2.3.1.13

Subtract $90 x^{2}$ from $- 90 x^{2}$ .

$y + C_{1} = \int 225 x^{4} - 180 x^{2} + 36 d x$

Step 2.3.2

Split the single integral into multiple integrals.

$y + C_{1} = \int 225 x^{4} d x + \int - 180 x^{2} d x + \int 36 d x$

Step 2.3.3

Since $225$ is constant with respect to $x$ , move $225$ out of the integral.

$y + C_{1} = 225 \int x^{4} d x + \int - 180 x^{2} d x + \int 36 d x$

Step 2.3.4

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$y + C_{1} = 225 (\frac{1}{5} x^{5} + C_{2}) + \int - 180 x^{2} d x + \int 36 d x$

Step 2.3.5

Since $- 180$ is constant with respect to $x$ , move $- 180$ out of the integral.

$y + C_{1} = 225 (\frac{1}{5} x^{5} + C_{2}) - 180 \int x^{2} d x + \int 36 d x$

Step 2.3.6

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

$y + C_{1} = 225 (\frac{1}{5} x^{5} + C_{2}) - 180 (\frac{1}{3} x^{3} + C_{3}) + \int 36 d x$

Step 2.3.7

Apply the constant rule.

$y + C_{1} = 225 (\frac{1}{5} x^{5} + C_{2}) - 180 (\frac{1}{3} x^{3} + C_{3}) + 36 x + C_{4}$

Step 2.3.8

Simplify.

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Step 2.3.8.1

Simplify.

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Step 2.3.8.1.1

Combine $\frac{1}{5}$ and $x^{5}$ .

$y + C_{1} = 225 (\frac{x^{5}}{5} + C_{2}) - 180 (\frac{1}{3} x^{3} + C_{3}) + 36 x + C_{4}$

Step 2.3.8.1.2

Combine $\frac{1}{3}$ and $x^{3}$ .

$y + C_{1} = 225 (\frac{x^{5}}{5} + C_{2}) - 180 (\frac{x^{3}}{3} + C_{3}) + 36 x + C_{4}$

Step 2.3.8.2

Simplify.

$y + C_{1} = 45 x^{5} - 60 x^{3} + 36 x + C_{5}$

Step 2.4

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

$y = 45 x^{5} - 60 x^{3} + 36 x + K$