Calculus Examples

$\int \frac{5}{9 x^{2} + 49} d x$

Step 1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int \frac{1}{9 x^{2} + 49} d x$

Step 2

Reorder $9 x^{2}$ and $49$ .

$5 \int \frac{1}{49 + 9 x^{2}} d x$

Step 3

Factor $9$ out of $49 + 9 x^{2}$ .

Tap for more steps...

Step 3.1

Factor $9$ out of $49$ .

$5 \int \frac{1}{9 (\frac{49}{9}) + 9 x^{2}} d x$

Step 3.2

Factor $9$ out of $9 x^{2}$ .

$5 \int \frac{1}{9 (\frac{49}{9}) + 9 (x^{2})} d x$

Step 3.3

Factor $9$ out of $9 (\frac{49}{9}) + 9 (x^{2})$ .

$5 \int \frac{1}{9 (\frac{49}{9} + x^{2})} d x$

Step 4

Since $\frac{1}{9}$ is constant with respect to $x$ , move $\frac{1}{9}$ out of the integral.

$5 (\frac{1}{9} \int \frac{1}{\frac{49}{9} + x^{2}} d x)$

Step 5

Combine $\frac{1}{9}$ and $5$ .

$\frac{5}{9} \int \frac{1}{\frac{49}{9} + x^{2}} d x$

Step 6

Rewrite $\frac{49}{9}$ as ${(\frac{7}{3})}^{2}$ .

$\frac{5}{9} \int \frac{1}{{(\frac{7}{3})}^{2} + x^{2}} d x$

Step 7

The integral of $\frac{1}{{(\frac{7}{3})}^{2} + x^{2}}$ with respect to $x$ is $\frac{1}{\frac{7}{3}} \arctan (\frac{x}{\frac{7}{3}}) + C$ .

$\frac{5}{9} (\frac{1}{\frac{7}{3}} \arctan (\frac{x}{\frac{7}{3}}) + C)$

Step 8

Simplify the answer.

Tap for more steps...

Step 8.1

Simplify.

Tap for more steps...

Step 8.1.1

Multiply by the reciprocal of the fraction to divide by $\frac{7}{3}$ .

$\frac{5}{9} (1 (\frac{3}{7}) \arctan (\frac{x}{\frac{7}{3}}) + C)$

Step 8.1.2

Multiply $\frac{3}{7}$ by $1$ .

$\frac{5}{9} (\frac{3}{7} \arctan (\frac{x}{\frac{7}{3}}) + C)$

Step 8.1.3

Multiply by the reciprocal of the fraction to divide by $\frac{7}{3}$ .

$\frac{5}{9} (\frac{3}{7} \arctan (x \frac{3}{7}) + C)$

Step 8.1.4

Combine $x$ and $\frac{3}{7}$ .

$\frac{5}{9} (\frac{3}{7} \arctan (\frac{x \cdot 3}{7}) + C)$

Step 8.1.5

Move $3$ to the left of $x$ .

$\frac{5}{9} (\frac{3}{7} \arctan (\frac{3 \cdot x}{7}) + C)$

Step 8.1.6

Combine $\frac{3}{7}$ and $\arctan (\frac{3 x}{7})$ .

$\frac{5}{9} (\frac{3 \arctan (\frac{3 x}{7})}{7} + C)$

Step 8.2

Rewrite $\frac{5}{9} (\frac{3 \arctan (\frac{3 x}{7})}{7} + C)$ as $\frac{5}{9} \cdot \frac{3}{7} \arctan (\frac{3}{7} x) + C$ .

$\frac{5}{9} \cdot \frac{3}{7} \arctan (\frac{3}{7} x) + C$

Step 8.3

Simplify.

Tap for more steps...

Step 8.3.1

Multiply $\frac{5}{9}$ by $\frac{3}{7}$ .

$\frac{5 \cdot 3}{9 \cdot 7} \arctan (\frac{3}{7} x) + C$

Step 8.3.2

Multiply $5$ by $3$ .

$\frac{15}{9 \cdot 7} \arctan (\frac{3}{7} x) + C$

Step 8.3.3

Multiply $9$ by $7$ .

$\frac{15}{63} \arctan (\frac{3}{7} x) + C$

Step 8.3.4

Cancel the common factor of $15$ and $63$ .

Tap for more steps...

Step 8.3.4.1

Factor $3$ out of $15$ .

$\frac{3 (5)}{63} \arctan (\frac{3}{7} x) + C$

Step 8.3.4.2

Cancel the common factors.

Tap for more steps...

Step 8.3.4.2.1

Factor $3$ out of $63$ .

$\frac{3 \cdot 5}{3 \cdot 21} \arctan (\frac{3}{7} x) + C$

Step 8.3.4.2.2

Cancel the common factor.

$\frac{3 \cdot 5}{3 \cdot 21} \arctan (\frac{3}{7} x) + C$

Step 8.3.4.2.3

Rewrite the expression.

$\frac{5}{21} \arctan (\frac{3}{7} x) + C$