Calculus Examples

$\int x yd x d x$

Step 1

Simplify.

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Step 1.1

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

$\int x^{1} x yd d x$

Step 1.2

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

$\int x^{1} x^{1} yd d x$

Step 1.3

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

$\int x^{1 + 1} yd d x$

Step 1.4

Add $1$ and $1$ .

$\int x^{2} yd d x$

Step 2

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

$yd \int x^{2} d x$

Step 3

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

$yd (\frac{1}{3} x^{3} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $yd (\frac{1}{3} x^{3} + C)$ as $yd \frac{1}{3} x^{3} + C$ .

$yd \frac{1}{3} x^{3} + C$

Step 4.2

Simplify.

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Step 4.2.1

Combine $yd$ and $\frac{1}{3}$ .

$\frac{1}{3} yd x^{3} + C$

Step 4.2.2

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

$\frac{x^{3}}{3} yd + C$

Step 4.3

Reorder terms.

$\frac{1}{3} x^{3} yd + C$