Calculus Examples

$\int \cos (x^{2}) \cdot 2 x d x$ , $u = x^{2}$

Step 1

Move $2$ to the left of $\cos (x^{2})$ .

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

Step 2

Let $u = x^{2}$ . Then $d u = 2 x d x$ . Rewrite using $u$ and $d u$ .

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Step 2.1

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

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Step 2.1.1

Differentiate $x^{2}$ .

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

Step 2.1.2

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

$2 x$

Step 2.2

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

$\int \cos (u) d u$

Step 3

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

$\sin (u) + C$

Step 4

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

$\sin (x^{2}) + C$

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