Calculus Examples

$\frac{d^{2} y}{d x^{2}} = \frac{21 \sec^{2} (x) \tan (x)}{10}$

Step 1

Integrate both sides with respect to $x$ .

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Step 1.1

The first derivative is equal to the integral of the second derivative with respect to $x$ .

$\frac{d y}{d x} = \int \frac{21 \sec^{2} (x) \tan (x)}{10} d x$

Step 1.2

Since $\frac{21}{10}$ is constant with respect to $x$ , move $\frac{21}{10}$ out of the integral.

$\frac{d y}{d x} = \frac{21}{10} \int \sec^{2} (x) \tan (x) d x$

Step 1.3

Let $u = \sec (x)$ . Then $d u = \sec (x) \tan (x) d x$ , so $\frac{1}{\sec (x) \tan (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.3.1

Let $u = \sec (x)$ . Find $\frac{d u}{d x}$ .

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Step 1.3.1.1

Differentiate $\sec (x)$ .

$\frac{d}{d x} [\sec (x)]$

Step 1.3.1.2

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$\sec (x) \tan (x)$

Step 1.3.2

Rewrite the problem using $u$ and $d u$ .

$\frac{d y}{d x} = \frac{21}{10} \int u d u$

Step 1.4

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

$\frac{d y}{d x} = \frac{21}{10} (\frac{1}{2} u^{2} + C_{1})$

Step 1.5

Simplify.

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Step 1.5.1

Rewrite $\frac{21}{10} (\frac{1}{2} u^{2} + C_{1})$ as $\frac{21}{10} \cdot \frac{1}{2} u^{2} + C_{1}$ .

$\frac{d y}{d x} = \frac{21}{10} \cdot \frac{1}{2} u^{2} + C_{1}$

Step 1.5.2

Simplify.

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Step 1.5.2.1

Multiply $\frac{21}{10}$ by $\frac{1}{2}$ .

$\frac{d y}{d x} = \frac{21}{10 \cdot 2} u^{2} + C_{1}$

Step 1.5.2.2

Multiply $10$ by $2$ .

$\frac{d y}{d x} = \frac{21}{20} u^{2} + C_{1}$

Step 1.6

Replace all occurrences of $u$ with $\sec (x)$ .

$\frac{d y}{d x} = \frac{21}{20} \sec^{2} (x) + C_{1}$

Step 2

Rewrite the equation.

$d y = (\frac{21}{20} \sec^{2} (x) + C_{1}) d x$

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

$\int d y = \int \frac{21}{20} \sec^{2} (x) + C_{1} d x$

Step 3.2

Apply the constant rule.

$y + C_{2} = \int \frac{21}{20} \sec^{2} (x) + C_{1} d x$

Step 3.3

Integrate the right side.

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Step 3.3.1

Split the single integral into multiple integrals.

$y + C_{2} = \int \frac{21}{20} \sec^{2} (x) d x + \int C_{1} d x$

Step 3.3.2

Since $\frac{21}{20}$ is constant with respect to $x$ , move $\frac{21}{20}$ out of the integral.

$y + C_{2} = \frac{21}{20} \int \sec^{2} (x) d x + \int C_{1} d x$

Step 3.3.3

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

$y + C_{2} = \frac{21}{20} (\tan (x) + C_{3}) + \int C_{1} d x$

Step 3.3.4

Apply the constant rule.

$y + C_{2} = \frac{21}{20} (\tan (x) + C_{3}) + C_{1} x + C_{4}$

Step 3.3.5

Simplify.

$y + C_{2} = \frac{21}{20} \tan (x) + C_{1} x + C_{5}$

Step 3.4

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

$y = \frac{21}{20} \tan (x) + C x + D$