Calculus Examples

$\frac{d y}{d x} - 2 x \ln (x^{2} + 1) = 0$

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d x}$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Simplify $- 2 \ln (x^{2} + 1)$ by moving $2$ inside the logarithm.

$\frac{d y}{d x} + x (- \ln ({(x^{2} + 1)}^{2})) = 0$

Step 1.1.1.2

Rewrite using the commutative property of multiplication.

$\frac{d y}{d x} - x \ln ({(x^{2} + 1)}^{2}) = 0$

Step 1.1.2

Add $x \ln ({(x^{2} + 1)}^{2})$ to both sides of the equation.

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

Step 1.2

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Rewrite $x \ln ({(x^{2} + 1)}^{2})$ as $x (2 \ln (x^{2} + 1))$ .

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

Step 2.3.2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 2.3.3

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x^{2} + 1)$ and $d v = x$ .

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

Step 2.3.4

Simplify.

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Step 2.3.4.1

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

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

Step 2.3.4.2

Combine $\ln (x^{2} + 1)$ and $\frac{x^{2}}{2}$ .

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

Step 2.3.4.3

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

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

Step 2.3.4.4

Multiply $\frac{x^{2}}{2}$ by $\frac{2 x}{x^{2} + 1}$ .

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

Step 2.3.4.5

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

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Step 2.3.4.5.1

Move $x$ .

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

Step 2.3.4.5.2

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

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Step 2.3.4.5.2.1

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

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

Step 2.3.4.5.2.2

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

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

Step 2.3.4.5.3

Add $1$ and $2$ .

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

Step 2.3.4.6

Move $2$ to the left of $x^{3}$ .

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

Step 2.3.4.7

Cancel the common factor of $2$ .

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Step 2.3.4.7.1

Cancel the common factor.

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

Step 2.3.4.7.2

Rewrite the expression.

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

Step 2.3.5

Divide $x^{3}$ by $x^{2} + 1$ .

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Step 2.3.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x^{2}$

$0 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 2.3.5.2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x^{2}$ .

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 2.3.5.3

Multiply the new quotient term by the divisor.

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$x$

Step 2.3.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 0 + x$

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$x$

Step 2.3.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$x$

Step 2.3.5.6

Pull the next term from the original dividend down into the current dividend.

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$x$

$0$

Step 2.3.5.7

The final answer is the quotient plus the remainder over the divisor.

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

Step 2.3.6

Split the single integral into multiple integrals.

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

Step 2.3.7

Move the negative in front of the fraction.

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

Step 2.3.8

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

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

Step 2.3.9

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

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

Step 2.3.10

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

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

Step 2.3.11

Let $u = x^{2} + 1$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.11.1

Let $u = x^{2} + 1$ . Find $\frac{d u}{d x}$ .

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Step 2.3.11.1.1

Differentiate $x^{2} + 1$ .

$\frac{d}{d x} [x^{2} + 1]$

Step 2.3.11.1.2

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$

Step 2.3.11.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [1]$

Step 2.3.11.1.4

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

$2 x + 0$

Step 2.3.11.1.5

Add $2 x$ and $0$ .

$2 x$

Step 2.3.11.2

Rewrite the problem using $u$ and $d u$ .

$y + C_{1} = 2 (\frac{\ln (x^{2} + 1) x^{2}}{2} - (\frac{x^{2}}{2} + C_{2} - \int \frac{1}{u} \cdot \frac{1}{2} d u))$

Step 2.3.12

Simplify.

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Step 2.3.12.1

Multiply $\frac{1}{u}$ by $\frac{1}{2}$ .

$y + C_{1} = 2 (\frac{\ln (x^{2} + 1) x^{2}}{2} - (\frac{x^{2}}{2} + C_{2} - \int \frac{1}{u \cdot 2} d u))$

Step 2.3.12.2

Move $2$ to the left of $u$ .

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

Step 2.3.13

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 2.3.14

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

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

Step 2.3.15

Simplify.

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Step 2.3.15.1

Simplify.

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

Step 2.3.15.2

Combine the numerators over the common denominator.

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

Step 2.3.16

Replace all occurrences of $u$ with $x^{2} + 1$ .

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

Step 2.3.17

Simplify.

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Step 2.3.17.1

Combine the numerators over the common denominator.

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

Step 2.3.17.2

Reorder factors in $\ln (x^{2} + 1) x^{2} - x^{2} + \ln (| x^{2} + 1 |)$ .

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

Step 2.3.17.3

Cancel the common factor of $2$ .

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Step 2.3.17.3.1

Cancel the common factor.

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

Step 2.3.17.3.2

Rewrite the expression.

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

Step 2.3.18

Reorder terms.

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

Step 2.4

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

$y = - x^{2} + x^{2} \ln (x^{2} + 1) + \ln (| x^{2} + 1 |) + C$