Calculus Examples

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

Step 1

Integrate both sides.

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Step 1.1

Set up an integral on each side.

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

Step 1.2

Integrate the left side.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

Rewrite $2 (\frac{1}{2} y^{2} + C_{1})$ as $2 (\frac{1}{2}) y^{2} + C_{1}$ .

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

Step 1.2.3.2

Simplify.

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Step 1.2.3.2.1

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

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

Step 1.2.3.2.2

Cancel the common factor of $2$ .

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Step 1.2.3.2.2.1

Cancel the common factor.

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

Step 1.2.3.2.2.2

Rewrite the expression.

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

Step 1.2.3.2.3

Multiply $y^{2}$ by $1$ .

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

Step 1.3

Integrate the right side.

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Step 1.3.1

Split the single integral into multiple integrals.

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

Step 1.3.2

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

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

Step 1.3.3

Apply the constant rule.

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

Step 1.3.4

Simplify.

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

Step 1.4

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

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

Step 2

Solve for $y$ .

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Step 2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

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

Step 2.2

Simplify $\pm \sqrt{\frac{1}{3} x^{3} + x + K}$ .

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Step 2.2.1

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

$y = \pm \sqrt{\frac{x^{3}}{3} + x + K}$

Step 2.2.2

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

$y = \pm \sqrt{\frac{x^{3}}{3} + x \cdot \frac{3}{3} + K}$

Step 2.2.3

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

$y = \pm \sqrt{\frac{x^{3}}{3} + \frac{x \cdot 3}{3} + K}$

Step 2.2.4

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x^{3} + x \cdot 3}{3} + K}$

Step 2.2.5

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

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Step 2.2.5.1

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

$y = \pm \sqrt{\frac{x \cdot x^{2} + x \cdot 3}{3} + K}$

Step 2.2.5.2

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

$y = \pm \sqrt{\frac{x \cdot x^{2} + x (3)}{3} + K}$

Step 2.2.5.3

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

$y = \pm \sqrt{\frac{x (x^{2} + 3)}{3} + K}$

Step 2.2.6

To write $K$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \pm \sqrt{\frac{x (x^{2} + 3)}{3} + K \cdot \frac{3}{3}}$

Step 2.2.7

Simplify terms.

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Step 2.2.7.1

Combine $K$ and $\frac{3}{3}$ .

$y = \pm \sqrt{\frac{x (x^{2} + 3)}{3} + \frac{K \cdot 3}{3}}$

Step 2.2.7.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x (x^{2} + 3) + K \cdot 3}{3}}$

Step 2.2.8

Simplify the numerator.

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Step 2.2.8.1

Apply the distributive property.

$y = \pm \sqrt{\frac{x \cdot x^{2} + x \cdot 3 + K \cdot 3}{3}}$

Step 2.2.8.2

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

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Step 2.2.8.2.1

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

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Step 2.2.8.2.1.1

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

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

Step 2.2.8.2.1.2

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

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

Step 2.2.8.2.2

Add $1$ and $2$ .

$y = \pm \sqrt{\frac{x^{3} + x \cdot 3 + K \cdot 3}{3}}$

Step 2.2.8.3

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

$y = \pm \sqrt{\frac{x^{3} + 3 \cdot x + K \cdot 3}{3}}$

Step 2.2.8.4

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

$y = \pm \sqrt{\frac{x^{3} + 3 x + 3 K}{3}}$

Step 2.2.9

Rewrite $\sqrt{\frac{x^{3} + 3 x + 3 K}{3}}$ as $\frac{\sqrt{x^{3} + 3 x + 3 K}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{x^{3} + 3 x + 3 K}}{\sqrt{3}}$

Step 2.2.10

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

$y = \pm \frac{\sqrt{x^{3} + 3 x + 3 K}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 2.2.11

Combine and simplify the denominator.

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Step 2.2.11.1

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

$y = \pm \frac{\sqrt{x^{3} + 3 x + 3 K} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.2.11.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 2.2.11.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 2.2.11.4

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

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

Step 2.2.11.5

Add $1$ and $1$ .

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

Step 2.2.11.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.2.11.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 2.2.11.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.11.6.3

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

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

Step 2.2.11.6.4

Cancel the common factor of $2$ .

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Step 2.2.11.6.4.1

Cancel the common factor.

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

Step 2.2.11.6.4.2

Rewrite the expression.

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

Step 2.2.11.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{x^{3} + 3 x + 3 K} \sqrt{3}}{3}$

Step 2.2.12

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(x^{3} + 3 x + 3 K) \cdot 3}}{3}$

Step 2.2.13

Reorder factors in $\pm \frac{\sqrt{(x^{3} + 3 x + 3 K) \cdot 3}}{3}$ .

$y = \pm \frac{\sqrt{3 (x^{3} + 3 x + 3 K)}}{3}$

Step 2.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{\sqrt{3 (x^{3} + 3 x + 3 K)}}{3}$

Step 2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{\sqrt{3 (x^{3} + 3 x + 3 K)}}{3}$

Step 2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{\sqrt{3 (x^{3} + 3 x + 3 K)}}{3}$

$y = - \frac{\sqrt{3 (x^{3} + 3 x + 3 K)}}{3}$

$y = \frac{\sqrt{3 (x^{3} + 3 x + 3 K)}}{3}$

$y = - \frac{\sqrt{3 (x^{3} + 3 x + 3 K)}}{3}$

$y = \frac{\sqrt{3 (x^{3} + 3 x + 3 K)}}{3}$

$y = - \frac{\sqrt{3 (x^{3} + 3 x + 3 K)}}{3}$

Step 3

Simplify the constant of integration.

$y = \frac{\sqrt{3 (x^{3} + 3 x + K)}}{3}$

$y = - \frac{\sqrt{3 (x^{3} + 3 x + K)}}{3}$