Math / Calculus

Question $\begin{array}{l}\int\left(\sec ^{2} x+7\right) d x \\ \int\left(\sec ^{2} x+7\right) d x=\square\end{array}$

Studdy Solution

STEP 1

1. We are given the integral $\int (\sec^2 x + 7) \, dx$ .
2. We need to find the antiderivative of the function $\sec^2 x + 7$ .

STEP 2

1. Break down the integral into simpler parts.
2. Integrate each part separately.
3. Combine the results to find the final integral.

STEP 3

Break down the integral into two separate integrals:
$\int (\sec^2 x + 7) \, dx = \int \sec^2 x \, dx + \int 7 \, dx$

STEP 4

Integrate the first part, $\int \sec^2 x \, dx$ :
The antiderivative of $\sec^2 x$ is $\tan x$ .
$\int \sec^2 x \, dx = \tan x + C_1$

STEP 5

Integrate the second part, $\int 7 \, dx$ :
The antiderivative of a constant $7$ is $7x$ .
$\int 7 \, dx = 7x + C_2$

STEP 6

Combine the results from the two integrals:
$\int (\sec^2 x + 7) \, dx = \tan x + 7x + C$
where $C = C_1 + C_2$ is the constant of integration.
The value of the integral is:
$\boxed{\tan x + 7x + C}$

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Question $\begin{array}{l}\int\left(\sec ^{2} x+7\right) d x \\ \int\left(\sec ^{2} x+7\right) d x=\square\end{array}$

Studdy Solution

STEP 1

1. We are given the integral $\int (\sec^2 x + 7) \, dx$ .
2. We need to find the antiderivative of the function $\sec^2 x + 7$ .

STEP 2

1. Break down the integral into simpler parts.
2. Integrate each part separately.
3. Combine the results to find the final integral.

STEP 3

Break down the integral into two separate integrals:
$\int (\sec^2 x + 7) \, dx = \int \sec^2 x \, dx + \int 7 \, dx$

STEP 4

Integrate the first part, $\int \sec^2 x \, dx$ :
The antiderivative of $\sec^2 x$ is $\tan x$ .
$\int \sec^2 x \, dx = \tan x + C_1$

STEP 5

Integrate the second part, $\int 7 \, dx$ :
The antiderivative of a constant $7$ is $7x$ .
$\int 7 \, dx = 7x + C_2$

STEP 6

Combine the results from the two integrals:
$\int (\sec^2 x + 7) \, dx = \tan x + 7x + C$
where $C = C_1 + C_2$ is the constant of integration.
The value of the integral is:
$\boxed{\tan x + 7x + C}$

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Question∫(sec⁡2x+7)dx∫(sec⁡2x+7)dx=□\begin{array}{l}\int\left(\sec ^{2} x+7\right) d x \\ \int\left(\sec ^{2} x+7\right) d x=\square\end{array}∫(sec2x+7)dx∫(sec2x+7)dx=□​

Studdy Solution

STEP 1

1. We are given the integral ∫(sec⁡2x+7) dx\int (\sec^2 x + 7) \, dx∫(sec2x+7)dx.2. We need to find the antiderivative of the function sec⁡2x+7\sec^2 x + 7sec2x+7.

STEP 2

1. Break down the integral into simpler parts.2. Integrate each part separately.3. Combine the results to find the final integral.

STEP 3

Break down the integral into two separate integrals:∫(sec⁡2x+7) dx=∫sec⁡2x dx+∫7 dx\int (\sec^2 x + 7) \, dx = \int \sec^2 x \, dx + \int 7 \, dx∫(sec2x+7)dx=∫sec2xdx+∫7dx

STEP 4

Integrate the first part, ∫sec⁡2x dx\int \sec^2 x \, dx∫sec2xdx:The antiderivative of sec⁡2x\sec^2 xsec2x is tan⁡x\tan xtanx.∫sec⁡2x dx=tan⁡x+C1\int \sec^2 x \, dx = \tan x + C_1∫sec2xdx=tanx+C1​

STEP 5

Integrate the second part, ∫7 dx\int 7 \, dx∫7dx:The antiderivative of a constant 777 is 7x7x7x.∫7 dx=7x+C2\int 7 \, dx = 7x + C_2∫7dx=7x+C2​

STEP 6

Combine the results from the two integrals:∫(sec⁡2x+7) dx=tan⁡x+7x+C\int (\sec^2 x + 7) \, dx = \tan x + 7x + C∫(sec2x+7)dx=tanx+7x+Cwhere C=C1+C2C = C_1 + C_2C=C1​+C2​ is the constant of integration. The value of the integral is:tan⁡x+7x+C\boxed{\tan x + 7x + C}tanx+7x+C​

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Question $\begin{array}{l}\int\left(\sec ^{2} x+7\right) d x \\ \int\left(\sec ^{2} x+7\right) d x=\square\end{array}$

1. We are given the integral $\int (\sec^2 x + 7) \, dx$ .
2. We need to find the antiderivative of the function $\sec^2 x + 7$ .

1. Break down the integral into simpler parts.
2. Integrate each part separately.
3. Combine the results to find the final integral.

Break down the integral into two separate integrals:
$\int (\sec^2 x + 7) \, dx = \int \sec^2 x \, dx + \int 7 \, dx$

Integrate the first part, $\int \sec^2 x \, dx$ :
The antiderivative of $\sec^2 x$ is $\tan x$ .
$\int \sec^2 x \, dx = \tan x + C_1$

Integrate the second part, $\int 7 \, dx$ :
The antiderivative of a constant $7$ is $7x$ .
$\int 7 \, dx = 7x + C_2$

Combine the results from the two integrals:
$\int (\sec^2 x + 7) \, dx = \tan x + 7x + C$
where $C = C_1 + C_2$ is the constant of integration.
The value of the integral is:
$\boxed{\tan x + 7x + C}$