Calculus Examples

$\int \frac{3}{4 x^{2} + 4} d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int \frac{1}{4 x^{2} + 4} d x$

Step 2

Reorder $4 x^{2}$ and $4$ .

$3 \int \frac{1}{4 + 4 x^{2}} d x$

Step 3

Factor $4$ out of $4 + 4 x^{2}$ .

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Step 3.1

Factor $4$ out of $4$ .

$3 \int \frac{1}{4 \cdot 1 + 4 x^{2}} d x$

Step 3.2

Factor $4$ out of $4 \cdot 1$ .

$3 \int \frac{1}{4 (1) + 4 x^{2}} d x$

Step 3.3

Factor $4$ out of $4 x^{2}$ .

$3 \int \frac{1}{4 (1) + 4 (x^{2})} d x$

Step 3.4

Factor $4$ out of $4 (1) + 4 (x^{2})$ .

$3 \int \frac{1}{4 (1 + x^{2})} d x$

Step 4

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$3 (\frac{1}{4} \int \frac{1}{1 + x^{2}} d x)$

Step 5

Combine fractions.

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Step 5.1

Combine $\frac{1}{4}$ and $3$ .

$\frac{3}{4} \int \frac{1}{1 + x^{2}} d x$

Step 5.2

Rewrite $1$ as $1^{2}$ .

$\frac{3}{4} \int \frac{1}{1^{2} + x^{2}} d x$

Step 6

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

$\frac{3}{4} (\arctan (x) + C)$

Step 7

Simplify.

$\frac{3}{4} \arctan (x) + C$