Calculus Examples

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

Step 1

Check if the left side of the equation is the result of the derivative of the term $(x + 1) y$ .

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Step 1.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x + 1$ and $g (x) = y$ .

$(x + 1) \frac{d}{d x} [y] + y \frac{d}{d x} [x + 1]$

Step 1.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$(x + 1) y' + y \frac{d}{d x} [x + 1]$

Step 1.3

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$(x + 1) y' + y (\frac{d}{d x} [x] + \frac{d}{d x} [1])$

Step 1.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$(x + 1) y' + y (1 + \frac{d}{d x} [1])$

Step 1.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$(x + 1) y' + y (1 + 0)$

Step 1.6

Add $1$ and $0$ .

$(x + 1) y' + y \cdot 1$

Step 1.7

Substitute $\frac{d y}{d x}$ for $y'$ .

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

Step 1.8

Reorder $y$ and $1$ .

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

Step 1.9

Multiply $y$ by $1$ .

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

Step 2

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [(x + 1) y] = x^{2} - 1$

Step 3

Set up an integral on each side.

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

Step 4

Integrate the left side.

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

Step 5

Integrate the right side.

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Step 5.1

Split the single integral into multiple integrals.

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

Step 5.2

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

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

Step 5.3

Apply the constant rule.

$(x + 1) y = \frac{1}{3} x^{3} + C - x + C$

Step 5.4

Simplify.

$(x + 1) y = \frac{1}{3} x^{3} - x + C$

Step 6

Divide each term in $(x + 1) y = \frac{1}{3} x^{3} - x + C$ by $x + 1$ and simplify.

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Step 6.1

Divide each term in $(x + 1) y = \frac{1}{3} x^{3} - x + C$ by $x + 1$ .

$\frac{(x + 1) y}{x + 1} = \frac{\frac{1}{3} x^{3}}{x + 1} + \frac{- x}{x + 1} + \frac{C}{x + 1}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $x + 1$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{(x + 1) y}{x + 1} = \frac{\frac{1}{3} x^{3}}{x + 1} + \frac{- x}{x + 1} + \frac{C}{x + 1}$

Step 6.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{3} x^{3}}{x + 1} + \frac{- x}{x + 1} + \frac{C}{x + 1}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

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

$y = \frac{\frac{x^{3}}{3}}{x + 1} + \frac{- x}{x + 1} + \frac{C}{x + 1}$

Step 6.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{x^{3}}{3} \cdot \frac{1}{x + 1} + \frac{- x}{x + 1} + \frac{C}{x + 1}$

Step 6.3.1.3

Multiply $\frac{x^{3}}{3}$ by $\frac{1}{x + 1}$ .

$y = \frac{x^{3}}{3 (x + 1)} + \frac{- x}{x + 1} + \frac{C}{x + 1}$

Step 6.3.1.4

Move the negative in front of the fraction.

$y = \frac{x^{3}}{3 (x + 1)} - \frac{x}{x + 1} + \frac{C}{x + 1}$

Step 6.3.2

To write $- \frac{x}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{x^{3}}{3 (x + 1)} - \frac{x}{x + 1} \cdot \frac{3}{3} + \frac{C}{x + 1}$

Step 6.3.3

Write each expression with a common denominator of $3 (x + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.3.3.1

Multiply $\frac{x}{x + 1}$ by $\frac{3}{3}$ .

$y = \frac{x^{3}}{3 (x + 1)} - \frac{x \cdot 3}{(x + 1) \cdot 3} + \frac{C}{x + 1}$

Step 6.3.3.2

Reorder the factors of $(x + 1) \cdot 3$ .

$y = \frac{x^{3}}{3 (x + 1)} - \frac{x \cdot 3}{3 (x + 1)} + \frac{C}{x + 1}$

Step 6.3.4

Combine the numerators over the common denominator.

$y = \frac{x^{3} - x \cdot 3}{3 (x + 1)} + \frac{C}{x + 1}$

Step 6.3.5

Simplify the numerator.

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Step 6.3.5.1

Factor $x$ out of $x^{3} - x \cdot 3$ .

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Step 6.3.5.1.1

Factor $x$ out of $x^{3}$ .

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

Step 6.3.5.1.2

Factor $x$ out of $- x \cdot 3$ .

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

Step 6.3.5.1.3

Factor $x$ out of $x \cdot x^{2} + x (- 1 \cdot 3)$ .

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

Step 6.3.5.2

Multiply $- 1$ by $3$ .

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

Step 6.3.6

To write $\frac{C}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 6.3.7

Write each expression with a common denominator of $3 (x + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.3.7.1

Multiply $\frac{C}{x + 1}$ by $\frac{3}{3}$ .

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

Step 6.3.7.2

Reorder the factors of $(x + 1) \cdot 3$ .

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

Step 6.3.8

Combine the numerators over the common denominator.

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

Step 6.3.9

Simplify the numerator.

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Step 6.3.9.1

Apply the distributive property.

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

Step 6.3.9.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 6.3.9.2.1

Multiply $x$ by $x^{2}$ .

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Step 6.3.9.2.1.1

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

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

Step 6.3.9.2.1.2

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

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

Step 6.3.9.2.2

Add $1$ and $2$ .

$y = \frac{x^{3} + x \cdot - 3 + C \cdot 3}{3 (x + 1)}$

Step 6.3.9.3

Move $- 3$ to the left of $x$ .

$y = \frac{x^{3} - 3 \cdot x + C \cdot 3}{3 (x + 1)}$

Step 6.3.9.4

Move $3$ to the left of $C$ .

$y = \frac{x^{3} - 3 x + 3 C}{3 (x + 1)}$