Calculus Examples

$\frac{d^{2} y}{d x^{2}} = (4.5) (\sec^{2} (x)) (\tan (x))$

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 $x$ .

$\frac{d y}{d x} = \int (4.5) (\sec^{2} (x)) (\tan (x)) d x$

Step 1.2

Multiply $4.5$ by $\sec^{2} (x)$ .

$\frac{d y}{d x} = \int 4.5 \sec^{2} (x) \tan (x) d x$

Step 1.3

Since $4.5$ is constant with respect to $x$ , move $4.5$ out of the integral.

$\frac{d y}{d x} = 4.5 \int \sec^{2} (x) \tan (x) d x$

Step 1.4

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 1.4.1

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

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Step 1.4.1.1

Differentiate $\sec (x)$ .

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

Step 1.4.1.2

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

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

Step 1.4.2

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

$\frac{d y}{d x} = 4.5 \int u d u$

Step 1.5

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

$\frac{d y}{d x} = 4.5 (\frac{1}{2} u^{2} + C_{1})$

Step 1.6

Simplify.

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Step 1.6.1

Rewrite $4.5 (\frac{1}{2} u^{2} + C_{1})$ as $4.5 (\frac{1}{2}) u^{2} + C_{1}$ .

$\frac{d y}{d x} = 4.5 (\frac{1}{2}) u^{2} + C_{1}$

Step 1.6.2

Combine $4.5$ and $\frac{1}{2}$ .

$\frac{d y}{d x} = \frac{4.5}{2} u^{2} + C_{1}$

Step 1.7

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

$\frac{d y}{d x} = \frac{4.5}{2} \sec^{2} (x) + C_{1}$

Step 1.8

Divide $4.5$ by $2$ .

$\frac{d y}{d x} = 2.25 \sec^{2} (x) + C_{1}$

Step 2

Rewrite the equation.

$d y = (2.25 \sec^{2} (x) + C_{1}) d x$

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

$\int d y = \int 2.25 \sec^{2} (x) + C_{1} d x$

Step 3.2

Apply the constant rule.

$y + C_{2} = \int 2.25 \sec^{2} (x) + C_{1} d x$

Step 3.3

Integrate the right side.

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Step 3.3.1

Split the single integral into multiple integrals.

$y + C_{2} = \int 2.25 \sec^{2} (x) d x + \int C_{1} d x$

Step 3.3.2

Since $2.25$ is constant with respect to $x$ , move $2.25$ out of the integral.

$y + C_{2} = 2.25 \int \sec^{2} (x) d x + \int C_{1} d x$

Step 3.3.3

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

$y + C_{2} = 2.25 (\tan (x) + C_{3}) + \int C_{1} d x$

Step 3.3.4

Apply the constant rule.

$y + C_{2} = 2.25 (\tan (x) + C_{3}) + C_{1} x + C_{4}$

Step 3.3.5

Simplify.

$y + C_{2} = 2.25 \tan (x) + C_{1} x + C_{5}$

Step 3.3.6

Reorder terms.

$y + C_{2} = C_{1} x + 2.25 \tan (x) + C_{5}$

Step 3.4

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

$y = C x + 2.25 \tan (x) + D$