Calculus Examples

$\int_{0}^{\frac{π}{3}} \frac{9 \sin (θ) + 9 \sin (θ) \tan^{2} (θ)}{\sec^{2} (θ)} d θ$

Step 1

Factor $9 \sin (θ)$ out of $9 \sin (θ)$ .

$\int_{0}^{\frac{π}{3}} \frac{9 \sin (θ) (1) + 9 \sin (θ) \tan^{2} (θ)}{\sec^{2} (θ)} d θ$

Step 2

Factor $9 \sin (θ)$ out of $9 \sin (θ) \tan^{2} (θ)$ .

$\int_{0}^{\frac{π}{3}} \frac{9 \sin (θ) (1) + 9 \sin (θ) (\tan^{2} (θ))}{\sec^{2} (θ)} d θ$

Step 3

Factor $9 \sin (θ)$ out of $9 \sin (θ) (1) + 9 \sin (θ) (\tan^{2} (θ))$ .

$\int_{0}^{\frac{π}{3}} \frac{9 \sin (θ) (1 + \tan^{2} (θ))}{\sec^{2} (θ)} d θ$

Step 4

Rearrange terms.

$\int_{0}^{\frac{π}{3}} \frac{9 \sin (θ) (\tan^{2} (θ) + 1)}{\sec^{2} (θ)} d θ$

Step 5

Apply pythagorean identity.

$\int_{0}^{\frac{π}{3}} \frac{9 \sin (θ) \sec^{2} (θ)}{\sec^{2} (θ)} d θ$

Step 6

Cancel the common factor of $\sec^{2} (θ)$ .

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Step 6.1

Cancel the common factor.

$\int_{0}^{\frac{π}{3}} \frac{9 \sin (θ) \sec^{2} (θ)}{\sec^{2} (θ)} d θ$

Step 6.2

Divide $9 \sin (θ)$ by $1$ .

$\int_{0}^{\frac{π}{3}} 9 \sin (θ) d θ$

Step 7

Since $9$ is constant with respect to $θ$ , move $9$ out of the integral.

$9 \int_{0}^{\frac{π}{3}} \sin (θ) d θ$

Step 8

The integral of $\sin (θ)$ with respect to $θ$ is $- \cos (θ)$ .

$9 (- \cos (θ)]_{0}^{\frac{π}{3}})$

Step 9

Simplify the answer.

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Step 9.1

Evaluate $- \cos (θ)$ at $\frac{π}{3}$ and at $0$ .

$9 (- \cos (\frac{π}{3}) + \cos (0))$

Step 9.2

Simplify.

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Step 9.2.1

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$9 (- \frac{1}{2} + \cos (0))$

Step 9.2.2

The exact value of $\cos (0)$ is $1$ .

$9 (- \frac{1}{2} + 1)$

Step 9.2.3

Write $1$ as a fraction with a common denominator.

$9 (- \frac{1}{2} + \frac{2}{2})$

Step 9.2.4

Combine the numerators over the common denominator.

$9 \frac{- 1 + 2}{2}$

Step 9.2.5

Add $- 1$ and $2$ .

$9 (\frac{1}{2})$

Step 9.2.6

Combine $9$ and $\frac{1}{2}$ .

$\frac{9}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{2}$

Decimal Form:

$4.5$

Mixed Number Form:

$4 \frac{1}{2}$