Calculus Examples

$\frac{d y}{d x} = \frac{1}{x \sqrt{49 x^{2} - 1}}$

Step 1

Rewrite the equation.

$d y = \frac{1}{x \sqrt{49 x^{2} - 1}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{1}{x \sqrt{49 x^{2} - 1}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{1}{x \sqrt{49 x^{2} - 1}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Simplify with factoring out.

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Step 2.3.1.1

Rewrite $49 x^{2}$ as ${(7 x)}^{2}$ .

$y + C_{1} = \int \frac{1}{x \sqrt{{(7 x)}^{2} - 1}} d x$

Step 2.3.1.2

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

$y + C_{1} = \int \frac{1}{x \sqrt{{(7 x)}^{2} - 1^{2}}} d x$

Step 2.3.2

The integral of $\frac{1}{x \sqrt{{(7 x)}^{2} - 1^{2}}}$ with respect to $x$ is $arcsec (| 7 x |)$

$y + C_{1} = arcsec (| 7 x |) + C_{2}$

Step 2.3.3

Rewrite $arcsec (| 7 x |) + C_{2}$ as $arcsec (7 | x |) + C_{2}$ .

$y + C_{1} = arcsec (7 | x |) + C_{2}$

Step 2.4

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

$y = arcsec (7 | x |) + K$