Calculus Examples

$\int \frac{\sqrt{2}}{3 t^{2} + 3} d t$

Step 1

Since $\sqrt{2}$ is constant with respect to $t$ , move $\sqrt{2}$ out of the integral.

$\sqrt{2} \int \frac{1}{3 t^{2} + 3} d t$

Step 2

Reorder $3 t^{2}$ and $3$ .

$\sqrt{2} \int \frac{1}{3 + 3 t^{2}} d t$

Step 3

Factor $3$ out of $3 + 3 t^{2}$ .

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Step 3.1

Factor $3$ out of $3$ .

$\sqrt{2} \int \frac{1}{3 \cdot 1 + 3 t^{2}} d t$

Step 3.2

Factor $3$ out of $3 \cdot 1$ .

$\sqrt{2} \int \frac{1}{3 (1) + 3 t^{2}} d t$

Step 3.3

Factor $3$ out of $3 t^{2}$ .

$\sqrt{2} \int \frac{1}{3 (1) + 3 (t^{2})} d t$

Step 3.4

Factor $3$ out of $3 (1) + 3 (t^{2})$ .

$\sqrt{2} \int \frac{1}{3 (1 + t^{2})} d t$

Step 4

Since $\frac{1}{3}$ is constant with respect to $t$ , move $\frac{1}{3}$ out of the integral.

$\sqrt{2} (\frac{1}{3} \int \frac{1}{1 + t^{2}} d t)$

Step 5

Combine fractions.

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Step 5.1

Combine $\frac{1}{3}$ and $\sqrt{2}$ .

$\frac{\sqrt{2}}{3} \int \frac{1}{1 + t^{2}} d t$

Step 5.2

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

$\frac{\sqrt{2}}{3} \int \frac{1}{1^{2} + t^{2}} d t$

Step 6

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

$\frac{\sqrt{2}}{3} (\arctan (t) + C)$

Step 7

Simplify.

$\frac{\sqrt{2}}{3} \arctan (t) + C$