Solved on Jan 31, 2024

Evaluate the indefinite integral $\int(3 - 12 \sec^2 4x) dx$ .

STEP 1

Assumptions
1. We need to integrate the function $3 - 12\sec^2(4x)$ with respect to $x$ .
2. The integral of a constant $a$ with respect to $x$ is $ax$ .
3. The integral of $\sec^2(u)$ with respect to $x$ is $\tan(u)$ , provided $u$ is a function of $x$ and $du/dx$ is a constant.

STEP 2

We will split the integral into two separate integrals to simplify the calculation.
$\int(3 - 12\sec^2(4x))dx = \int 3dx - \int 12\sec^2(4x)dx$

STEP 3

First, we integrate the constant $3$ with respect to $x$ .
$\int 3dx = 3x$

STEP 4

Next, we need to integrate $-12\sec^2(4x)$ with respect to $x$ . We notice that the inner function is $4x$ , so we will need to use a substitution to solve this integral.

STEP 5

Let $u = 4x$ . Then, $\frac{du}{dx} = 4$ , which means $dx = \frac{du}{4}$ .

STEP 6

Substitute $u$ and $dx$ into the integral.
$\int 12\sec^2(4x)dx = \int 12\sec^2(u) \frac{du}{4}$

STEP 7

Simplify the integral by taking the constant $12/4$ outside the integral.
$\int 12\sec^2(u) \frac{du}{4} = 3\int \sec^2(u) du$

STEP 8

Now, integrate $\sec^2(u)$ with respect to $u$ .
$\int \sec^2(u) du = \tan(u)$

STEP 9

Substitute back the value of $u$ to get the integral in terms of $x$ .
$\tan(u) = \tan(4x)$

STEP 10

Combine the results from the integrals of the constant and the $\sec^2(4x)$ term.
$3x - 3\tan(4x)$

STEP 11

Add the constant of integration $C$ to the result, as the integral of a function is indefinite.
$3x - 3\tan(4x) + C$

STEP 12

Write the final answer.
$\int(3 - 12\sec^2(4x))dx = 3x - 3\tan(4x) + C$
The integral of $3 - 12\sec^2(4x)$ with respect to $x$ is $3x - 3\tan(4x) + C$ .

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Evaluate the indefinite integral $\int(3 - 12 \sec^2 4x) dx$ .

STEP 1

Assumptions
1. We need to integrate the function $3 - 12\sec^2(4x)$ with respect to $x$ .
2. The integral of a constant $a$ with respect to $x$ is $ax$ .
3. The integral of $\sec^2(u)$ with respect to $x$ is $\tan(u)$ , provided $u$ is a function of $x$ and $du/dx$ is a constant.

STEP 2

We will split the integral into two separate integrals to simplify the calculation.
$\int(3 - 12\sec^2(4x))dx = \int 3dx - \int 12\sec^2(4x)dx$

STEP 3

First, we integrate the constant $3$ with respect to $x$ .
$\int 3dx = 3x$

STEP 4

Next, we need to integrate $-12\sec^2(4x)$ with respect to $x$ . We notice that the inner function is $4x$ , so we will need to use a substitution to solve this integral.

STEP 5

Let $u = 4x$ . Then, $\frac{du}{dx} = 4$ , which means $dx = \frac{du}{4}$ .

STEP 6

Substitute $u$ and $dx$ into the integral.
$\int 12\sec^2(4x)dx = \int 12\sec^2(u) \frac{du}{4}$

STEP 7

Simplify the integral by taking the constant $12/4$ outside the integral.
$\int 12\sec^2(u) \frac{du}{4} = 3\int \sec^2(u) du$

STEP 8

Now, integrate $\sec^2(u)$ with respect to $u$ .
$\int \sec^2(u) du = \tan(u)$

STEP 9

Substitute back the value of $u$ to get the integral in terms of $x$ .
$\tan(u) = \tan(4x)$

STEP 10

Combine the results from the integrals of the constant and the $\sec^2(4x)$ term.
$3x - 3\tan(4x)$

STEP 11

Add the constant of integration $C$ to the result, as the integral of a function is indefinite.
$3x - 3\tan(4x) + C$

STEP 12

Write the final answer.
$\int(3 - 12\sec^2(4x))dx = 3x - 3\tan(4x) + C$
The integral of $3 - 12\sec^2(4x)$ with respect to $x$ is $3x - 3\tan(4x) + C$ .

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Evaluate the indefinite integral ∫(3−12sec⁡24x)dx\int(3 - 12 \sec^2 4x) dx∫(3−12sec24x)dx.

STEP 1

STEP 2

We will split the integral into two separate integrals to simplify the calculation.∫(3−12sec⁡2(4x))dx=∫3dx−∫12sec⁡2(4x)dx\int(3 - 12\sec^2(4x))dx = \int 3dx - \int 12\sec^2(4x)dx∫(3−12sec2(4x))dx=∫3dx−∫12sec2(4x)dx

STEP 3

First, we integrate the constant 333 with respect to xxx.∫3dx=3x\int 3dx = 3x∫3dx=3x

STEP 4

Next, we need to integrate −12sec⁡2(4x)-12\sec^2(4x)−12sec2(4x) with respect to xxx. We notice that the inner function is 4x4x4x, so we will need to use a substitution to solve this integral.

STEP 5

Let u=4xu = 4xu=4x. Then, dudx=4\frac{du}{dx} = 4dxdu​=4, which means dx=du4dx = \frac{du}{4}dx=4du​.

STEP 6

Substitute uuu and dxdxdx into the integral.∫12sec⁡2(4x)dx=∫12sec⁡2(u)du4\int 12\sec^2(4x)dx = \int 12\sec^2(u) \frac{du}{4}∫12sec2(4x)dx=∫12sec2(u)4du​

STEP 7

Simplify the integral by taking the constant 12/412/412/4 outside the integral.∫12sec⁡2(u)du4=3∫sec⁡2(u)du\int 12\sec^2(u) \frac{du}{4} = 3\int \sec^2(u) du∫12sec2(u)4du​=3∫sec2(u)du

STEP 8

Now, integrate sec⁡2(u)\sec^2(u)sec2(u) with respect to uuu.∫sec⁡2(u)du=tan⁡(u)\int \sec^2(u) du = \tan(u)∫sec2(u)du=tan(u)

STEP 9

Substitute back the value of uuu to get the integral in terms of xxx.tan⁡(u)=tan⁡(4x)\tan(u) = \tan(4x)tan(u)=tan(4x)

STEP 10

Combine the results from the integrals of the constant and the sec⁡2(4x)\sec^2(4x)sec2(4x) term.3x−3tan⁡(4x)3x - 3\tan(4x)3x−3tan(4x)

STEP 11

Add the constant of integration CCC to the result, as the integral of a function is indefinite.3x−3tan⁡(4x)+C3x - 3\tan(4x) + C3x−3tan(4x)+C

STEP 12