Calculus Examples

$\frac{d v}{d t} = \frac{6}{t \sqrt{t^{2} - 1}}$

Step 1

Rewrite the equation.

$d v = \frac{6}{t \sqrt{t^{2} - 1}} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d v = \int \frac{6}{t \sqrt{t^{2} - 1}} d t$

Step 2.2

Apply the constant rule.

$v + C_{1} = \int \frac{6}{t \sqrt{t^{2} - 1}} d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $6$ is constant with respect to $t$ , move $6$ out of the integral.

$v + C_{1} = 6 \int \frac{1}{t \sqrt{t^{2} - 1}} d t$

Step 2.3.2

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

$v + C_{1} = 6 \int \frac{1}{t \sqrt{t^{2} - 1^{2}}} d t$

Step 2.3.3

The integral of $\frac{1}{t \sqrt{t^{2} - 1^{2}}}$ with respect to $t$ is $arcsec (| t |)$

$v + C_{1} = 6 (arcsec (| t |) + C_{2})$

Step 2.3.4

Simplify.

$v + C_{1} = 6 arcsec (| t |) + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$v = 6 arcsec (| t |) + K$