Calculus Examples

$\int \frac{44}{x^{2} \sqrt{x^{2} + 16}} d x$

Step 1

Since $44$ is constant with respect to $x$ , move $44$ out of the integral.

$44 \int \frac{1}{x^{2} \sqrt{x^{2} + 16}} d x$

Step 2

Let $x = 4 \tan (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = 4 \sec^{2} (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $4 \sec^{2} (t)$ is positive.

$44 \int \frac{1}{{(4 \tan (t))}^{2} \sqrt{{(4 \tan (t))}^{2} + 16}} (4 \sec^{2} (t)) d t$

Step 3

Simplify terms.

Tap for more steps...

Step 3.1

Simplify $\sqrt{{(4 \tan (t))}^{2} + 16}$ .

Tap for more steps...

Step 3.1.1

Simplify each term.

Tap for more steps...

Step 3.1.1.1

Apply the product rule to $4 \tan (t)$ .

$44 \int \frac{1}{{(4 \tan (t))}^{2} \sqrt{4^{2} \tan^{2} (t) + 16}} (4 \sec^{2} (t)) d t$

Step 3.1.1.2

Raise $4$ to the power of $2$ .

$44 \int \frac{1}{{(4 \tan (t))}^{2} \sqrt{16 \tan^{2} (t) + 16}} (4 \sec^{2} (t)) d t$

Step 3.1.2

Factor $16$ out of $16 \tan^{2} (t) + 16$ .

Tap for more steps...

Step 3.1.2.1

Factor $16$ out of $16 \tan^{2} (t)$ .

$44 \int \frac{1}{{(4 \tan (t))}^{2} \sqrt{16 (\tan^{2} (t)) + 16}} (4 \sec^{2} (t)) d t$

Step 3.1.2.2

Factor $16$ out of $16$ .

$44 \int \frac{1}{{(4 \tan (t))}^{2} \sqrt{16 (\tan^{2} (t)) + 16 (1)}} (4 \sec^{2} (t)) d t$

Step 3.1.2.3

Factor $16$ out of $16 (\tan^{2} (t)) + 16 (1)$ .

$44 \int \frac{1}{{(4 \tan (t))}^{2} \sqrt{16 (\tan^{2} (t) + 1)}} (4 \sec^{2} (t)) d t$

Step 3.1.3

Apply pythagorean identity.

$44 \int \frac{1}{{(4 \tan (t))}^{2} \sqrt{16 \sec^{2} (t)}} (4 \sec^{2} (t)) d t$

Step 3.1.4

Rewrite $16 \sec^{2} (t)$ as ${(4 \sec (t))}^{2}$ .

$44 \int \frac{1}{{(4 \tan (t))}^{2} \sqrt{{(4 \sec (t))}^{2}}} (4 \sec^{2} (t)) d t$

Step 3.1.5

Pull terms out from under the radical, assuming positive real numbers.

$44 \int \frac{1}{{(4 \tan (t))}^{2} (4 \sec (t))} (4 \sec^{2} (t)) d t$

Step 3.2

Simplify terms.

Tap for more steps...

Step 3.2.1

Cancel the common factor of $4 \sec (t)$ .

Tap for more steps...

Step 3.2.1.1

Factor $4 \sec (t)$ out of ${(4 \tan (t))}^{2} (4 \sec (t))$ .

$44 \int \frac{1}{4 \sec (t) ({(4 \tan (t))}^{2})} (4 \sec^{2} (t)) d t$

Step 3.2.1.2

Factor $4 \sec (t)$ out of $4 \sec^{2} (t)$ .

$44 \int \frac{1}{4 \sec (t) ({(4 \tan (t))}^{2})} (4 \sec (t) (\sec (t))) d t$

Step 3.2.1.3

Cancel the common factor.

$44 \int \frac{1}{4 \sec (t) {(4 \tan (t))}^{2}} (4 \sec (t) \sec (t)) d t$

Step 3.2.1.4

Rewrite the expression.

$44 \int \frac{1}{{(4 \tan (t))}^{2}} \sec (t) d t$

Step 3.2.2

Combine $\frac{1}{{(4 \tan (t))}^{2}}$ and $\sec (t)$ .

$44 \int \frac{\sec (t)}{{(4 \tan (t))}^{2}} d t$

Step 3.2.3

Simplify.

Tap for more steps...

Step 3.2.3.1

Factor $4$ out of $4 \tan (t)$ .

$44 \int \frac{\sec (t)}{{(4 (\tan (t)))}^{2}} d t$

Step 3.2.3.2

Apply the product rule to $4 (\tan (t))$ .

$44 \int \frac{\sec (t)}{4^{2} \tan^{2} (t)} d t$

Step 3.2.3.3

Raise $4$ to the power of $2$ .

$44 \int \frac{\sec (t)}{16 \tan^{2} (t)} d t$

Step 4

Since $\frac{1}{16}$ is constant with respect to $t$ , move $\frac{1}{16}$ out of the integral.

$44 (\frac{1}{16} \int \frac{\sec (t)}{\tan^{2} (t)} d t)$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Combine $\frac{1}{16}$ and $44$ .

$\frac{44}{16} \int \frac{\sec (t)}{\tan^{2} (t)} d t$

Step 5.2

Cancel the common factor of $44$ and $16$ .

Tap for more steps...

Step 5.2.1

Factor $4$ out of $44$ .

$\frac{4 (11)}{16} \int \frac{\sec (t)}{\tan^{2} (t)} d t$

Step 5.2.2

Cancel the common factors.

Tap for more steps...

Step 5.2.2.1

Factor $4$ out of $16$ .

$\frac{4 \cdot 11}{4 \cdot 4} \int \frac{\sec (t)}{\tan^{2} (t)} d t$

Step 5.2.2.2

Cancel the common factor.

$\frac{4 \cdot 11}{4 \cdot 4} \int \frac{\sec (t)}{\tan^{2} (t)} d t$

Step 5.2.2.3

Rewrite the expression.

$\frac{11}{4} \int \frac{\sec (t)}{\tan^{2} (t)} d t$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Rewrite $\sec (x)$ as $\frac{\csc (x)}{\cot (x)}$ .

$\frac{11}{4} \int \frac{\frac{\csc (t)}{\cot (t)}}{\tan^{2} (t)} d t$

Step 6.2

Apply the reciprocal identity.

$\frac{11}{4} \int \frac{\frac{\csc (t)}{\cot (t)}}{{(\frac{1}{\cot (t)})}^{2}} d t$

Step 6.3

Simplify.

Tap for more steps...

Step 6.3.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{11}{4} \int \frac{\csc (t)}{\cot (t)} \cdot \frac{1}{{(\frac{1}{\cot (t)})}^{2}} d t$

Step 6.3.2

Combine.

$\frac{11}{4} \int \frac{\csc (t) \cdot 1}{\cot (t) {(\frac{1}{\cot (t)})}^{2}} d t$

Step 6.3.3

Multiply $\csc (t)$ by $1$ .

$\frac{11}{4} \int \frac{\csc (t)}{\cot (t) {(\frac{1}{\cot (t)})}^{2}} d t$

Step 6.3.4

Simplify the denominator.

Tap for more steps...

Step 6.3.4.1

Apply the product rule to $\frac{1}{\cot (t)}$ .

$\frac{11}{4} \int \frac{\csc (t)}{\cot (t) \frac{1^{2}}{{\cot (t)}^{2}}} d t$

Step 6.3.4.2

One to any power is one.

$\frac{11}{4} \int \frac{\csc (t)}{\cot (t) \frac{1}{{\cot (t)}^{2}}} d t$

Step 6.3.5

Combine $\cot (t)$ and $\frac{1}{{\cot (t)}^{2}}$ .

$\frac{11}{4} \int \frac{\csc (t)}{\frac{\cot (t)}{{\cot (t)}^{2}}} d t$

Step 6.3.6

Reduce the expression $\frac{\cot (t)}{{\cot (t)}^{2}}$ by cancelling the common factors.

Tap for more steps...

Step 6.3.6.1

Multiply by $1$ .

$\frac{11}{4} \int \frac{\csc (t)}{\frac{\cot (t) \cdot 1}{{\cot (t)}^{2}}} d t$

Step 6.3.6.2

Factor $\cot (t)$ out of ${\cot (t)}^{2}$ .

$\frac{11}{4} \int \frac{\csc (t)}{\frac{\cot (t) \cdot 1}{\cot (t) \cot (t)}} d t$

Step 6.3.6.3

Cancel the common factor.

$\frac{11}{4} \int \frac{\csc (t)}{\frac{\cot (t) \cdot 1}{\cot (t) \cot (t)}} d t$

Step 6.3.6.4

Rewrite the expression.

$\frac{11}{4} \int \frac{\csc (t)}{\frac{1}{\cot (t)}} d t$

Step 6.3.7

Multiply the numerator by the reciprocal of the denominator.

$\frac{11}{4} \int \csc (t) \cot (t) d t$

Step 7

Since the derivative of $- \csc (t)$ is $\csc (t) \cot (t)$ , the integral of $\csc (t) \cot (t)$ is $- \csc (t)$ .

$\frac{11}{4} (- \csc (t) + C)$

Step 8

Simplify.

Tap for more steps...

Step 8.1

Simplify.

$\frac{11}{4} (- \csc (t)) + C$

Step 8.2

Combine $\frac{11}{4}$ and $\csc (t)$ .

$- \frac{11 \csc (t)}{4} + C$

Step 9

Replace all occurrences of $t$ with $\arctan (\frac{x}{4})$ .

$- \frac{11 \csc (\arctan (\frac{x}{4}))}{4} + C$

Step 10

Simplify.

Tap for more steps...

Step 10.1

Simplify the numerator.

Tap for more steps...

Step 10.1.1

Draw a triangle in the plane with vertices $(1, \frac{x}{4})$ , $(1, 0)$ , and the origin. Then $\arctan (\frac{x}{4})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \frac{x}{4})$ . Therefore, $\csc (\arctan (\frac{x}{4}))$ is $\frac{\sqrt{1 + {(\frac{x}{4})}^{2}}}{\frac{x}{4}}$ .

$- \frac{11 \frac{\sqrt{1 + {(\frac{x}{4})}^{2}}}{\frac{x}{4}}}{4} + C$

Step 10.1.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{11 (\sqrt{1 + {(\frac{x}{4})}^{2}} \frac{4}{x})}{4} + C$

Step 10.1.3

Apply the product rule to $\frac{x}{4}$ .

$- \frac{11 (\sqrt{1 + \frac{x^{2}}{4^{2}}} \frac{4}{x})}{4} + C$

Step 10.1.4

Raise $4$ to the power of $2$ .

$- \frac{11 (\sqrt{1 + \frac{x^{2}}{16}} \frac{4}{x})}{4} + C$

Step 10.1.5

Write $1$ as a fraction with a common denominator.

$- \frac{11 (\sqrt{\frac{16}{16} + \frac{x^{2}}{16}} \frac{4}{x})}{4} + C$

Step 10.1.6

Combine the numerators over the common denominator.

$- \frac{11 (\sqrt{\frac{16 + x^{2}}{16}} \frac{4}{x})}{4} + C$

Step 10.1.7

Rewrite $\frac{16 + x^{2}}{16}$ as ${(\frac{1}{4})}^{2} (16 + x^{2})$ .

Tap for more steps...

Step 10.1.7.1

Factor the perfect power $1^{2}$ out of $16 + x^{2}$ .

$- \frac{11 (\sqrt{\frac{1^{2} (16 + x^{2})}{16}} \frac{4}{x})}{4} + C$

Step 10.1.7.2

Factor the perfect power $4^{2}$ out of $16$ .

$- \frac{11 (\sqrt{\frac{1^{2} (16 + x^{2})}{4^{2} \cdot 1}} \frac{4}{x})}{4} + C$

Step 10.1.7.3

Rearrange the fraction $\frac{1^{2} (16 + x^{2})}{4^{2} \cdot 1}$ .

$- \frac{11 (\sqrt{{(\frac{1}{4})}^{2} (16 + x^{2})} \frac{4}{x})}{4} + C$

Step 10.1.8

Pull terms out from under the radical.

$- \frac{11 (\frac{1}{4} \sqrt{16 + x^{2}} \frac{4}{x})}{4} + C$

Step 10.1.9

Combine $\frac{1}{4}$ and $\sqrt{16 + x^{2}}$ .

$- \frac{11 (\frac{\sqrt{16 + x^{2}}}{4} \cdot \frac{4}{x})}{4} + C$

Step 10.1.10

Combine.

$- \frac{11 \frac{\sqrt{16 + x^{2}} \cdot 4}{4 x}}{4} + C$

Step 10.1.11

Cancel the common factor of $4$ .

Tap for more steps...

Step 10.1.11.1

Cancel the common factor.

$- \frac{11 \frac{\sqrt{16 + x^{2}} \cdot 4}{4 x}}{4} + C$

Step 10.1.11.2

Rewrite the expression.

$- \frac{11 \frac{\sqrt{16 + x^{2}}}{x}}{4} + C$

Step 10.2

Combine $11$ and $\frac{\sqrt{16 + x^{2}}}{x}$ .

$- \frac{\frac{11 \sqrt{16 + x^{2}}}{x}}{4} + C$

Step 10.3

Multiply the numerator by the reciprocal of the denominator.

$- (\frac{11 \sqrt{16 + x^{2}}}{x} \cdot \frac{1}{4}) + C$

Step 10.4

Combine.

$- \frac{11 \sqrt{16 + x^{2}} \cdot 1}{x \cdot 4} + C$

Step 10.5

Multiply $11$ by $1$ .

$- \frac{11 \sqrt{16 + x^{2}}}{x \cdot 4} + C$

Step 10.6

Move $4$ to the left of $x$ .

$- \frac{11 \sqrt{16 + x^{2}}}{4 x} + C$