Calculus Examples

$2 \sec^{2} (2 x) - \frac{1}{x^{2}}$

Step 1

Write $2 \sec^{2} (2 x) - \frac{1}{x^{2}}$ as a function.

$f (x) = 2 \sec^{2} (2 x) - \frac{1}{x^{2}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int 2 \sec^{2} (2 x) - \frac{1}{x^{2}} d x$

Step 4

Split the single integral into multiple integrals.

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

Step 5

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 6

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 6.1

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

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Step 6.1.1

Differentiate $2 x$ .

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

Step 6.1.2

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 6.1.3

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 6.1.4

Multiply $2$ by $1$ .

$2$

Step 6.2

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

$2 \int \sec^{2} (u) \frac{1}{2} d u + \int - \frac{1}{x^{2}} d x$

Step 7

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

$2 \int \frac{\sec^{2} (u)}{2} d u + \int - \frac{1}{x^{2}} d x$

Step 8

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

$2 (\frac{1}{2} \int \sec^{2} (u) d u) + \int - \frac{1}{x^{2}} d x$

Step 9

Simplify.

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Step 9.1

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

$\frac{2}{2} \int \sec^{2} (u) d u + \int - \frac{1}{x^{2}} d x$

Step 9.2

Cancel the common factor of $2$ .

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Step 9.2.1

Cancel the common factor.

$\frac{2}{2} \int \sec^{2} (u) d u + \int - \frac{1}{x^{2}} d x$

Step 9.2.2

Rewrite the expression.

$1 \int \sec^{2} (u) d u + \int - \frac{1}{x^{2}} d x$

Step 9.3

Multiply $\int \sec^{2} (u) d u$ by $1$ .

$\int \sec^{2} (u) d u + \int - \frac{1}{x^{2}} d x$

Step 10

Since the derivative of $\tan (u)$ is $\sec^{2} (u)$ , the integral of $\sec^{2} (u)$ is $\tan (u)$ .

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

Step 11

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

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

Step 12

Apply basic rules of exponents.

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Step 12.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$\tan (u) + C - \int {(x^{2})}^{- 1} d x$

Step 12.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 12.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\tan (u) + C - \int x^{2 \cdot - 1} d x$

Step 12.2.2

Multiply $2$ by $- 1$ .

$\tan (u) + C - \int x^{- 2} d x$

Step 13

By the Power Rule, the integral of $x^{- 2}$ with respect to $x$ is $- x^{- 1}$ .

$\tan (u) + C - (- x^{- 1} + C)$

Step 14

Simplify.

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Step 14.1

Simplify.

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

Step 14.2

Simplify.

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Step 14.2.1

Multiply $- 1$ by $- 1$ .

$\tan (u) + 1 \frac{1}{x} + C$

Step 14.2.2

Multiply $\frac{1}{x}$ by $1$ .

$\tan (u) + \frac{1}{x} + C$

Step 15

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

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

Step 16

The answer is the antiderivative of the function $f (x) = 2 \sec^{2} (2 x) - \frac{1}{x^{2}}$ .

$F (x) =$ $\tan (2 x) + \frac{1}{x} + C$