Calculus Examples

$\cos (x) (\sec (x) - \cos (x))$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$\int \cos (x) \sec (x) + \cos (x) (- \cos (x)) d x$

Step 1.2

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.2.1

Reorder $\cos (x)$ and $\sec (x)$ .

$\int \sec (x) \cos (x) + \cos (x) (- \cos (x)) d x$

Step 1.2.2

Rewrite $\cos (x) \sec (x)$ in terms of sines and cosines.

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

Step 1.2.3

Cancel the common factors.

$\int 1 + \cos (x) (- \cos (x)) d x$

Step 1.3

Rewrite using the commutative property of multiplication.

$\int 1 - \cos (x) \cos (x) d x$

Step 1.4

Multiply $- \cos (x) \cos (x)$ .

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Step 1.4.1

Raise $\cos (x)$ to the power of $1$ .

$\int 1 - (\cos^{1} (x) \cos (x)) d x$

Step 1.4.2

Raise $\cos (x)$ to the power of $1$ .

$\int 1 - (\cos^{1} (x) \cos^{1} (x)) d x$

Step 1.4.3

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

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

Step 1.4.4

Add $1$ and $1$ .

$\int 1 - \cos^{2} (x) d x$

Step 1.5

Apply pythagorean identity.

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

Step 2

Use the half-angle formula to rewrite $\sin^{2} (x)$ as $\frac{1 - \cos (2 x)}{2}$ .

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

Step 3

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

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

Apply the constant rule.

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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Step 7.1.1

Differentiate $2 x$ .

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

Step 7.1.2

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

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

Step 7.1.3

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

$2 \cdot 1$

Step 7.1.4

Multiply $2$ by $1$ .

$2$

Step 7.2

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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

Step 12

Replace all occurrences of $u$ with $2 x$ .

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

Step 13

Simplify.

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Step 13.1

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

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

Step 13.2

Apply the distributive property.

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

Step 13.3

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

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

Step 13.4

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

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Step 13.4.1

Multiply $\frac{1}{2}$ by $\frac{\sin (2 x)}{2}$ .

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

Step 13.4.2

Multiply $2$ by $2$ .

$\frac{x}{2} - \frac{\sin (2 x)}{4} + C$

Step 14

Reorder terms.

$\frac{1}{2} x - \frac{1}{4} \sin (2 x) + C$