Math / Calculus

QuestionEvaluate the following antiderivatives. a) $\int 8 \cos (x)+\sec ^{2}(x) d x=\square+C$

Studdy Solution

STEP 1

Assumptions
1. We are given the integral $\int (8 \cos(x) + \sec^2(x)) \, dx$ .
2. We need to find the antiderivative of the given function.
3. The constant of integration is denoted by $C$ .

STEP 2

The integral can be split into two separate integrals:
$\int (8 \cos(x) + \sec^2(x)) \, dx = \int 8 \cos(x) \, dx + \int \sec^2(x) \, dx$

STEP 3

Evaluate the first integral $\int 8 \cos(x) \, dx$ .
The antiderivative of $\cos(x)$ is $\sin(x)$ , so:
$\int 8 \cos(x) \, dx = 8 \int \cos(x) \, dx = 8 \sin(x)$

STEP 4

Evaluate the second integral $\int \sec^2(x) \, dx$ .
The antiderivative of $\sec^2(x)$ is $\tan(x)$ , so:
$\int \sec^2(x) \, dx = \tan(x)$

STEP 5

Combine the results of the two integrals and add the constant of integration $C$ :
$\int (8 \cos(x) + \sec^2(x)) \, dx = 8 \sin(x) + \tan(x) + C$
The antiderivative is $8 \sin(x) + \tan(x) + C$ .

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QuestionEvaluate the following antiderivatives. a) $\int 8 \cos (x)+\sec ^{2}(x) d x=\square+C$

Studdy Solution

STEP 1

Assumptions
1. We are given the integral $\int (8 \cos(x) + \sec^2(x)) \, dx$ .
2. We need to find the antiderivative of the given function.
3. The constant of integration is denoted by $C$ .

STEP 2

The integral can be split into two separate integrals:
$\int (8 \cos(x) + \sec^2(x)) \, dx = \int 8 \cos(x) \, dx + \int \sec^2(x) \, dx$

STEP 3

Evaluate the first integral $\int 8 \cos(x) \, dx$ .
The antiderivative of $\cos(x)$ is $\sin(x)$ , so:
$\int 8 \cos(x) \, dx = 8 \int \cos(x) \, dx = 8 \sin(x)$

STEP 4

Evaluate the second integral $\int \sec^2(x) \, dx$ .
The antiderivative of $\sec^2(x)$ is $\tan(x)$ , so:
$\int \sec^2(x) \, dx = \tan(x)$

STEP 5

Combine the results of the two integrals and add the constant of integration $C$ :
$\int (8 \cos(x) + \sec^2(x)) \, dx = 8 \sin(x) + \tan(x) + C$
The antiderivative is $8 \sin(x) + \tan(x) + C$ .

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QuestionEvaluate the following antiderivatives. a) ∫8cos⁡(x)+sec⁡2(x)dx=□+C\int 8 \cos (x)+\sec ^{2}(x) d x=\square+C∫8cos(x)+sec2(x)dx=□+C

Studdy Solution

STEP 1

Assumptions1. We are given the integral ∫(8cos⁡(x)+sec⁡2(x)) dx\int (8 \cos(x) + \sec^2(x)) \, dx∫(8cos(x)+sec2(x))dx.2. We need to find the antiderivative of the given function.3. The constant of integration is denoted by CCC.

STEP 2

The integral can be split into two separate integrals:∫(8cos⁡(x)+sec⁡2(x)) dx=∫8cos⁡(x) dx+∫sec⁡2(x) dx\int (8 \cos(x) + \sec^2(x)) \, dx = \int 8 \cos(x) \, dx + \int \sec^2(x) \, dx∫(8cos(x)+sec2(x))dx=∫8cos(x)dx+∫sec2(x)dx

STEP 3

Evaluate the first integral ∫8cos⁡(x) dx\int 8 \cos(x) \, dx∫8cos(x)dx.The antiderivative of cos⁡(x)\cos(x)cos(x) is sin⁡(x)\sin(x)sin(x), so:∫8cos⁡(x) dx=8∫cos⁡(x) dx=8sin⁡(x)\int 8 \cos(x) \, dx = 8 \int \cos(x) \, dx = 8 \sin(x)∫8cos(x)dx=8∫cos(x)dx=8sin(x)

STEP 4

Evaluate the second integral ∫sec⁡2(x) dx\int \sec^2(x) \, dx∫sec2(x)dx.The antiderivative of sec⁡2(x)\sec^2(x)sec2(x) is tan⁡(x)\tan(x)tan(x), so:∫sec⁡2(x) dx=tan⁡(x)\int \sec^2(x) \, dx = \tan(x)∫sec2(x)dx=tan(x)

STEP 5

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QuestionEvaluate the following antiderivatives. a) $\int 8 \cos (x)+\sec ^{2}(x) d x=\square+C$

Assumptions
1. We are given the integral $\int (8 \cos(x) + \sec^2(x)) \, dx$ .
2. We need to find the antiderivative of the given function.
3. The constant of integration is denoted by $C$ .

The integral can be split into two separate integrals:
$\int (8 \cos(x) + \sec^2(x)) \, dx = \int 8 \cos(x) \, dx + \int \sec^2(x) \, dx$

Evaluate the first integral $\int 8 \cos(x) \, dx$ .
The antiderivative of $\cos(x)$ is $\sin(x)$ , so:
$\int 8 \cos(x) \, dx = 8 \int \cos(x) \, dx = 8 \sin(x)$

Evaluate the second integral $\int \sec^2(x) \, dx$ .
The antiderivative of $\sec^2(x)$ is $\tan(x)$ , so:
$\int \sec^2(x) \, dx = \tan(x)$