Calculus Examples

$\frac{d y}{d θ} = \sqrt{θ} - \sec^{2} (θ)$

Step 1

Rewrite the equation.

$d y = (\sqrt{θ} - \sec^{2} (θ)) d θ$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \sqrt{θ} - \sec^{2} (θ) d θ$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \sqrt{θ} - \sec^{2} (θ) d θ$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int \sqrt{θ} d θ + \int - \sec^{2} (θ) d θ$

Step 2.3.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{θ}$ as $θ^{\frac{1}{2}}$ .

$y + C_{1} = \int θ^{\frac{1}{2}} d θ + \int - \sec^{2} (θ) d θ$

Step 2.3.3

By the Power Rule, the integral of $θ^{\frac{1}{2}}$ with respect to $θ$ is $\frac{2}{3} θ^{\frac{3}{2}}$ .

$y + C_{1} = \frac{2}{3} θ^{\frac{3}{2}} + C_{2} + \int - \sec^{2} (θ) d θ$

Step 2.3.4

Since $- 1$ is constant with respect to $θ$ , move $- 1$ out of the integral.

$y + C_{1} = \frac{2}{3} θ^{\frac{3}{2}} + C_{2} - \int \sec^{2} (θ) d θ$

Step 2.3.5

Since the derivative of $\tan (θ)$ is $\sec^{2} (θ)$ , the integral of $\sec^{2} (θ)$ is $\tan (θ)$ .

$y + C_{1} = \frac{2}{3} θ^{\frac{3}{2}} + C_{2} - (\tan (θ) + C_{3})$

Step 2.3.6

Simplify.

$y + C_{1} = \frac{2}{3} θ^{\frac{3}{2}} - \tan (θ) + C_{4}$

Step 2.4

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

$y = \frac{2}{3} θ^{\frac{3}{2}} - \tan (θ) + K$