Calculus Examples

$\frac{d y}{d x} = (x + 1) (x + 2)$

Step 1

Rewrite the equation.

$d y = (x + 1) (x + 2) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int (x + 1) (x + 2) d x$

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Simplify.

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Step 2.3.1.1

Apply the distributive property.

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

Step 2.3.1.2

Apply the distributive property.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.1.4

Reorder $x$ and $2$ .

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

Step 2.3.1.5

Raise $x$ to the power of $1$ .

$y + C_{1} = \int x^{1} x + 2 x + 1 x + 1 \cdot 2 d x$

Step 2.3.1.6

Raise $x$ to the power of $1$ .

$y + C_{1} = \int x^{1} x^{1} + 2 x + 1 x + 1 \cdot 2 d x$

Step 2.3.1.7

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

$y + C_{1} = \int x^{1 + 1} + 2 x + 1 x + 1 \cdot 2 d x$

Step 2.3.1.8

Add $1$ and $1$ .

$y + C_{1} = \int x^{2} + 2 x + 1 x + 1 \cdot 2 d x$

Step 2.3.1.9

Multiply $x$ by $1$ .

$y + C_{1} = \int x^{2} + 2 x + x + 1 \cdot 2 d x$

Step 2.3.1.10

Multiply $2$ by $1$ .

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

Step 2.3.1.11

Add $2 x$ and $x$ .

$y + C_{1} = \int x^{2} + 3 x + 2 d x$

Step 2.3.2

Split the single integral into multiple integrals.

$y + C_{1} = \int x^{2} d x + \int 3 x d x + \int 2 d x$

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

Apply the constant rule.

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

Step 2.3.7

Simplify.

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Step 2.3.7.1

Combine $\frac{1}{2}$ and $x^{2}$ .

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

Step 2.3.7.2

Simplify.

$y + C_{1} = \frac{x^{3}}{3} + \frac{3 x^{2}}{2} + 2 x + C_{5}$

Step 2.3.7.3

Reorder terms.

$y + C_{1} = \frac{1}{3} x^{3} + \frac{3}{2} x^{2} + 2 x + C_{5}$

Step 2.4

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

$y = \frac{1}{3} x^{3} + \frac{3}{2} x^{2} + 2 x + K$