Math / Calculus

QuestionFind the indefinite integral. (Use $C$ for the constant of integration.) $\int \frac{9}{\sqrt{81-(x+9)^{2}}} d x$

Studdy Solution

STEP 1

1. We are given the integral $\int \frac{9}{\sqrt{81-(x+9)^{2}}} \, dx$ .
2. We need to find the indefinite integral and include the constant of integration $C$ .

STEP 2

1. Recognize the form of the integral.
2. Use a trigonometric substitution to simplify the integral.
3. Simplify the expression using the substitution.
4. Integrate the simplified expression.
5. Substitute back to the original variable.
6. Include the constant of integration.

STEP 3

Recognize that the integral is of the form $\int \frac{a}{\sqrt{a^2 - u^2}} \, du$ , which is related to the arcsine function.

STEP 4

Use the substitution $u = x + 9$ , then $du = dx$ .
The integral becomes:
$\int \frac{9}{\sqrt{81 - u^2}} \, du$

STEP 5

Recognize that $81 = 9^2$ , so the integral becomes:
$\int \frac{9}{\sqrt{9^2 - u^2}} \, du$
This is in the standard form for the arcsine function.

STEP 6

The integral $\int \frac{1}{\sqrt{a^2 - u^2}} \, du$ is $\arcsin\left(\frac{u}{a}\right) + C$ .
Thus, our integral becomes:
$9 \cdot \arcsin\left(\frac{u}{9}\right) + C$

STEP 7

Substitute back $u = x + 9$ :
$9 \cdot \arcsin\left(\frac{x + 9}{9}\right) + C$

STEP 8

Include the constant of integration $C$ .
The indefinite integral is:
$\boxed{9 \arcsin\left(\frac{x + 9}{9}\right) + C}$

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QuestionFind the indefinite integral. (Use $C$ for the constant of integration.) $\int \frac{9}{\sqrt{81-(x+9)^{2}}} d x$

Studdy Solution

STEP 1

1. We are given the integral $\int \frac{9}{\sqrt{81-(x+9)^{2}}} \, dx$ .
2. We need to find the indefinite integral and include the constant of integration $C$ .

STEP 2

1. Recognize the form of the integral.
2. Use a trigonometric substitution to simplify the integral.
3. Simplify the expression using the substitution.
4. Integrate the simplified expression.
5. Substitute back to the original variable.
6. Include the constant of integration.

STEP 3

Recognize that the integral is of the form $\int \frac{a}{\sqrt{a^2 - u^2}} \, du$ , which is related to the arcsine function.

STEP 4

Use the substitution $u = x + 9$ , then $du = dx$ .
The integral becomes:
$\int \frac{9}{\sqrt{81 - u^2}} \, du$

STEP 5

Recognize that $81 = 9^2$ , so the integral becomes:
$\int \frac{9}{\sqrt{9^2 - u^2}} \, du$
This is in the standard form for the arcsine function.

STEP 6

The integral $\int \frac{1}{\sqrt{a^2 - u^2}} \, du$ is $\arcsin\left(\frac{u}{a}\right) + C$ .
Thus, our integral becomes:
$9 \cdot \arcsin\left(\frac{u}{9}\right) + C$

STEP 7

Substitute back $u = x + 9$ :
$9 \cdot \arcsin\left(\frac{x + 9}{9}\right) + C$

STEP 8

Include the constant of integration $C$ .
The indefinite integral is:
$\boxed{9 \arcsin\left(\frac{x + 9}{9}\right) + C}$

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QuestionFind the indefinite integral. (Use CCC for the constant of integration.) ∫981−(x+9)2dx\int \frac{9}{\sqrt{81-(x+9)^{2}}} d x∫81−(x+9)2​9​dx

Studdy Solution

STEP 1

1. We are given the integral ∫981−(x+9)2 dx\int \frac{9}{\sqrt{81-(x+9)^{2}}} \, dx∫81−(x+9)2​9​dx.2. We need to find the indefinite integral and include the constant of integration CCC.

STEP 2

1. Recognize the form of the integral.2. Use a trigonometric substitution to simplify the integral.3. Simplify the expression using the substitution.4. Integrate the simplified expression.5. Substitute back to the original variable.6. Include the constant of integration.

STEP 3

Recognize that the integral is of the form ∫aa2−u2 du\int \frac{a}{\sqrt{a^2 - u^2}} \, du∫a2−u2​a​du, which is related to the arcsine function.

STEP 4

Use the substitution u=x+9 u = x + 9 u=x+9, then du=dx du = dx du=dx.The integral becomes:∫981−u2 du \int \frac{9}{\sqrt{81 - u^2}} \, du ∫81−u2​9​du

STEP 5

Recognize that 81=92 81 = 9^2 81=92, so the integral becomes:∫992−u2 du \int \frac{9}{\sqrt{9^2 - u^2}} \, du ∫92−u2​9​duThis is in the standard form for the arcsine function.

STEP 6

The integral ∫1a2−u2 du\int \frac{1}{\sqrt{a^2 - u^2}} \, du∫a2−u2​1​du is arcsin⁡(ua)+C\arcsin\left(\frac{u}{a}\right) + Carcsin(au​)+C.Thus, our integral becomes:9⋅arcsin⁡(u9)+C 9 \cdot \arcsin\left(\frac{u}{9}\right) + C 9⋅arcsin(9u​)+C

STEP 7

Substitute back u=x+9 u = x + 9 u=x+9:9⋅arcsin⁡(x+99)+C 9 \cdot \arcsin\left(\frac{x + 9}{9}\right) + C 9⋅arcsin(9x+9​)+C

STEP 8

Include the constant of integration CCC.The indefinite integral is:9arcsin⁡(x+99)+C \boxed{9 \arcsin\left(\frac{x + 9}{9}\right) + C} 9arcsin(9x+9​)+C​

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QuestionFind the indefinite integral. (Use $C$ for the constant of integration.) $\int \frac{9}{\sqrt{81-(x+9)^{2}}} d x$

1. We are given the integral $\int \frac{9}{\sqrt{81-(x+9)^{2}}} \, dx$ .
2. We need to find the indefinite integral and include the constant of integration $C$ .

1. Recognize the form of the integral.
2. Use a trigonometric substitution to simplify the integral.
3. Simplify the expression using the substitution.
4. Integrate the simplified expression.
5. Substitute back to the original variable.
6. Include the constant of integration.

Recognize that the integral is of the form $\int \frac{a}{\sqrt{a^2 - u^2}} \, du$ , which is related to the arcsine function.

Use the substitution $u = x + 9$ , then $du = dx$ .
The integral becomes:
$\int \frac{9}{\sqrt{81 - u^2}} \, du$

Recognize that $81 = 9^2$ , so the integral becomes:
$\int \frac{9}{\sqrt{9^2 - u^2}} \, du$
This is in the standard form for the arcsine function.

The integral $\int \frac{1}{\sqrt{a^2 - u^2}} \, du$ is $\arcsin\left(\frac{u}{a}\right) + C$ .
Thus, our integral becomes:
$9 \cdot \arcsin\left(\frac{u}{9}\right) + C$

Substitute back $u = x + 9$ :
$9 \cdot \arcsin\left(\frac{x + 9}{9}\right) + C$

Include the constant of integration $C$ .
The indefinite integral is:
$\boxed{9 \arcsin\left(\frac{x + 9}{9}\right) + C}$