Calculus Examples

$\int x^{5} \cos (x^{3}) d x$

Step 1

Let $u = x^{3}$ . Then $d u = 3 x^{2} d x$ , so $\frac{1}{3} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x^{3}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{3}$ .

$\frac{d}{d x} [x^{3}]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2}$

Step 1.2

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

$\int u \cos (u) \frac{1}{3} d u$

Step 2

Simplify.

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Step 2.1

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

$\int \frac{u}{3} \cos (u) d u$

Step 2.2

Combine $\frac{u}{3}$ and $\cos (u)$ .

$\int \frac{u \cos (u)}{3} d u$

Step 3

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int u \cos (u) d u$

Step 4

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

$\frac{1}{3} (u \sin (u) - \int \sin (u) d u)$

Step 5

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

$\frac{1}{3} (u \sin (u) - (- \cos (u) + C))$

Step 6

Rewrite $\frac{1}{3} (u \sin (u) - (- \cos (u) + C))$ as $\frac{1}{3} (u \sin (u) + \cos (u)) + C$ .

$\frac{1}{3} (u \sin (u) + \cos (u)) + C$

Step 7

Replace all occurrences of $u$ with $x^{3}$ .

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

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

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

Step 8.2

Multiply $\frac{1}{3} (x^{3} \sin (x^{3}))$ .

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Step 8.2.1

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

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

Step 8.2.2

Combine $\frac{x^{3}}{3}$ and $\sin (x^{3})$ .

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

Step 8.3

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

$\frac{x^{3} \sin (x^{3})}{3} + \frac{\cos (x^{3})}{3} + C$

Step 9

Reorder terms.

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