Calculus Examples

$x \sin (x d x)$

Step 1

Simplify.

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Step 1.1

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

$\int x \sin (d (x^{1} x)) d x$

Step 1.2

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

$\int x \sin (d (x^{1} x^{1})) d x$

Step 1.3

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

$\int x \sin (d x^{1 + 1}) d x$

Step 1.4

Add $1$ and $1$ .

$\int x \sin (d x^{2}) d x$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $d x^{2}$ .

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

Step 2.1.2

Since $d$ is constant with respect to $x$ , the derivative of $d x^{2}$ with respect to $x$ is $d \frac{d}{d x} [x^{2}]$ .

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

Step 2.1.3

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

$d (2 x)$

Step 2.1.4

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

$2 d x$

Step 2.1.5

Reorder $2 d$ and $x$ .

$x (2 d)$

Step 2.1.6

Reorder $x$ and $2$ .

$2 \cdot x d$

Step 2.2

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

$\int \sin (u) \frac{1}{2 d} d u$

Step 3

Combine $\sin (u)$ and $\frac{1}{2 d}$ .

$\int \frac{\sin (u)}{2 d} d u$

Step 4

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

$\frac{1}{2 d} \int \sin (u) d u$

Step 5

The integral of $\sin (u)$ with respect to $u$ is $- \cos (u)$ .

$\frac{1}{2 d} (- \cos (u) + C)$

Step 6

Simplify.

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Step 6.1

Simplify.

$\frac{1}{2 d} (- \cos (u)) + C$

Step 6.2

Combine $\frac{1}{2 d}$ and $\cos (u)$ .

$- \frac{\cos (u)}{2 d} + C$

Step 7

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

$- \frac{\cos (d x^{2})}{2 d} + C$