Calculus Examples

$\frac{d y}{d x} = x^{4} \sqrt{x^{3}}$ , $y (0) = 0$

Step 1

Rewrite the equation.

$d y = x^{4} \sqrt{x^{3}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int x^{4} \sqrt{x^{3}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int x^{4} \sqrt{x^{3}} d x$

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

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

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

Step 2.3.1.2

Multiply $x^{4}$ by $x^{\frac{3}{2}}$ by adding the exponents.

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Step 2.3.1.2.1

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

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

Step 2.3.1.2.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.3.1.2.3

Combine $4$ and $\frac{2}{2}$ .

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

Step 2.3.1.2.4

Combine the numerators over the common denominator.

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

Step 2.3.1.2.5

Simplify the numerator.

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Step 2.3.1.2.5.1

Multiply $4$ by $2$ .

$y + C_{1} = \int x^{\frac{8 + 3}{2}} d x$

Step 2.3.1.2.5.2

Add $8$ and $3$ .

$y + C_{1} = \int x^{\frac{11}{2}} d x$

Step 2.3.2

By the Power Rule, the integral of $x^{\frac{11}{2}}$ with respect to $x$ is $\frac{2}{13} x^{\frac{13}{2}}$ .

$y + C_{1} = \frac{2}{13} x^{\frac{13}{2}} + C_{2}$

Step 2.4

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

$y = \frac{2}{13} x^{\frac{13}{2}} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $0$ for $y$ in $y = \frac{2}{13} x^{\frac{13}{2}} + K$ .

$0 = \frac{2}{13} \cdot 0^{\frac{13}{2}} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $\frac{2}{13} \cdot 0^{\frac{13}{2}} + K = 0$ .

$\frac{2}{13} \cdot 0^{\frac{13}{2}} + K = 0$

Step 4.2

Simplify $\frac{2}{13} \cdot 0^{\frac{13}{2}} + K$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Rewrite $0$ as $0^{2}$ .

$\frac{2}{13} \cdot {(0^{2})}^{\frac{13}{2}} + K = 0$

Step 4.2.1.2

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

$\frac{2}{13} \cdot 0^{2 (\frac{13}{2})} + K = 0$

Step 4.2.1.3

Cancel the common factor of $2$ .

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Step 4.2.1.3.1

Cancel the common factor.

$\frac{2}{13} \cdot 0^{2 (\frac{13}{2})} + K = 0$

Step 4.2.1.3.2

Rewrite the expression.

$\frac{2}{13} \cdot 0^{13} + K = 0$

Step 4.2.1.4

Raising $0$ to any positive power yields $0$ .

$\frac{2}{13} \cdot 0 + K = 0$

Step 4.2.1.5

Multiply $\frac{2}{13}$ by $0$ .

$0 + K = 0$

Step 4.2.2

Add $0$ and $K$ .

$K = 0$

Step 5

Substitute $0$ for $K$ in $y = \frac{2}{13} x^{\frac{13}{2}} + K$ and simplify.

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Step 5.1

Substitute $0$ for $K$ .

$y = \frac{2}{13} x^{\frac{13}{2}} + 0$

Step 5.2

Add $\frac{2}{13} x^{\frac{13}{2}}$ and $0$ .

$y = \frac{2}{13} x^{\frac{13}{2}}$

Step 5.3

Combine $\frac{2}{13}$ and $x^{\frac{13}{2}}$ .

$y = \frac{2 x^{\frac{13}{2}}}{13}$