Calculus Examples

$\int x^{3} \sin (x^{2}) d x$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $x^{2}$ .

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

Step 1.1.2

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 1.2

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

Integrate by parts using the formula $\int w d v = w v - \int v d w$ , where $w = u$ and $dv = \sin (u)$ .

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

Step 5

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

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

Step 6

Simplify.

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Step 6.1

Multiply $- 1$ by $- 1$ .

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

Step 6.2

Multiply $\int \cos (u) d u$ by $1$ .

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

Step 7

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

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

Step 8

Rewrite $\frac{1}{2} (u (- \cos (u)) + \sin (u) + C)$ as $\frac{1}{2} (- u \cos (u) + \sin (u)) + C$ .

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

Step 9

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

$\frac{1}{2} (- x^{2} \cos (x^{2}) + \sin (x^{2})) + C$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$\frac{1}{2} (- x^{2} \cos (x^{2})) + \frac{1}{2} \sin (x^{2}) + C$

Step 10.2

Multiply $\frac{1}{2} (- x^{2} \cos (x^{2}))$ .

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Step 10.2.1

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

$- \frac{x^{2}}{2} \cos (x^{2}) + \frac{1}{2} \sin (x^{2}) + C$

Step 10.2.2

Combine $\cos (x^{2})$ and $\frac{x^{2}}{2}$ .

$- \frac{\cos (x^{2}) x^{2}}{2} + \frac{1}{2} \sin (x^{2}) + C$

Step 10.3

Combine $\frac{1}{2}$ and $\sin (x^{2})$ .

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

Step 10.4

Reorder factors in $- \frac{\cos (x^{2}) x^{2}}{2} + \frac{\sin (x^{2})}{2}$ .

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

Step 11

Reorder terms.

$- \frac{1}{2} x^{2} \cos (x^{2}) + \frac{1}{2} \sin (x^{2}) + C$