Math / Calculus

QuestionFind the following antiderivatives:
1. $\int 6+\sec ^{2}(x) d x=\square+C$ .
2. $\int \frac{7 x^{2}-6}{x^{2}} d x=\square+C$ .

Studdy Solution

STEP 1

1. We are finding indefinite integrals, which include a constant of integration $C$ .
2. We will use basic integration rules and properties to solve the integrals.

STEP 2

1. Solve the first integral: $\int (6 + \sec^2(x)) \, dx$ .
2. Solve the second integral: $\int \frac{7x^2 - 6}{x^2} \, dx$ .

STEP 3

To solve the first integral $\int (6 + \sec^2(x)) \, dx$ , we use the linearity of integration, which allows us to integrate each term separately:
$\int (6 + \sec^2(x)) \, dx = \int 6 \, dx + \int \sec^2(x) \, dx$

STEP 4

Integrate each term separately:
1. The integral of a constant $6$ is $6x$ .
2. The integral of $\sec^2(x)$ is $\tan(x)$ .
So, we have:
$\int 6 \, dx = 6x$ $\int \sec^2(x) \, dx = \tan(x)$

STEP 5

Combine the results and add the constant of integration $C$ :
$\int (6 + \sec^2(x)) \, dx = 6x + \tan(x) + C$

STEP 6

To solve the second integral $\int \frac{7x^2 - 6}{x^2} \, dx$ , first simplify the integrand by dividing each term by $x^2$ :
$\frac{7x^2 - 6}{x^2} = \frac{7x^2}{x^2} - \frac{6}{x^2} = 7 - 6x^{-2}$
So, the integral becomes:
$\int (7 - 6x^{-2}) \, dx$

STEP 7

Integrate each term separately:
1. The integral of $7$ is $7x$ .
2. The integral of $-6x^{-2}$ is $6x^{-1}$ (since the integral of $x^n$ is $\frac{x^{n+1}}{n+1}$ for $n \neq -1$ ).
So, we have:
$\int 7 \, dx = 7x$ $\int -6x^{-2} \, dx = 6x^{-1} = \frac{6}{x}$

STEP 8

Combine the results and add the constant of integration $C$ :
$\int \frac{7x^2 - 6}{x^2} \, dx = 7x + \frac{6}{x} + C$
The solutions to the antiderivatives are:
1. $\int 6+\sec ^{2}(x) \, dx = 6x + \tan(x) + C$
2. $\int \frac{7 x^{2}-6}{x^{2}} \, dx = 7x + \frac{6}{x} + C$

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QuestionFind the following antiderivatives:
1. $\int 6+\sec ^{2}(x) d x=\square+C$ .
2. $\int \frac{7 x^{2}-6}{x^{2}} d x=\square+C$ .

Studdy Solution

STEP 1

1. We are finding indefinite integrals, which include a constant of integration $C$ .
2. We will use basic integration rules and properties to solve the integrals.

STEP 2

1. Solve the first integral: $\int (6 + \sec^2(x)) \, dx$ .
2. Solve the second integral: $\int \frac{7x^2 - 6}{x^2} \, dx$ .

STEP 3

To solve the first integral $\int (6 + \sec^2(x)) \, dx$ , we use the linearity of integration, which allows us to integrate each term separately:
$\int (6 + \sec^2(x)) \, dx = \int 6 \, dx + \int \sec^2(x) \, dx$

STEP 4

Integrate each term separately:
1. The integral of a constant $6$ is $6x$ .
2. The integral of $\sec^2(x)$ is $\tan(x)$ .
So, we have:
$\int 6 \, dx = 6x$ $\int \sec^2(x) \, dx = \tan(x)$

STEP 5

Combine the results and add the constant of integration $C$ :
$\int (6 + \sec^2(x)) \, dx = 6x + \tan(x) + C$

STEP 6

To solve the second integral $\int \frac{7x^2 - 6}{x^2} \, dx$ , first simplify the integrand by dividing each term by $x^2$ :
$\frac{7x^2 - 6}{x^2} = \frac{7x^2}{x^2} - \frac{6}{x^2} = 7 - 6x^{-2}$
So, the integral becomes:
$\int (7 - 6x^{-2}) \, dx$

STEP 7

Integrate each term separately:
1. The integral of $7$ is $7x$ .
2. The integral of $-6x^{-2}$ is $6x^{-1}$ (since the integral of $x^n$ is $\frac{x^{n+1}}{n+1}$ for $n \neq -1$ ).
So, we have:
$\int 7 \, dx = 7x$ $\int -6x^{-2} \, dx = 6x^{-1} = \frac{6}{x}$

STEP 8

Combine the results and add the constant of integration $C$ :
$\int \frac{7x^2 - 6}{x^2} \, dx = 7x + \frac{6}{x} + C$
The solutions to the antiderivatives are:
1. $\int 6+\sec ^{2}(x) \, dx = 6x + \tan(x) + C$
2. $\int \frac{7 x^{2}-6}{x^{2}} \, dx = 7x + \frac{6}{x} + C$

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QuestionFind the following antiderivatives:1. ∫6+sec⁡2(x)dx=□+C\int 6+\sec ^{2}(x) d x=\square+C∫6+sec2(x)dx=□+C.2. ∫7x2−6x2dx=□+C\int \frac{7 x^{2}-6}{x^{2}} d x=\square+C∫x27x2−6​dx=□+C.

Studdy Solution

STEP 1

1. We are finding indefinite integrals, which include a constant of integration C C C.2. We will use basic integration rules and properties to solve the integrals.

STEP 2

1. Solve the first integral: ∫(6+sec⁡2(x)) dx\int (6 + \sec^2(x)) \, dx∫(6+sec2(x))dx.2. Solve the second integral: ∫7x2−6x2 dx\int \frac{7x^2 - 6}{x^2} \, dx∫x27x2−6​dx.

STEP 3

STEP 4

Integrate each term separately:1. The integral of a constant 6 6 6 is 6x 6x 6x.2. The integral of sec⁡2(x)\sec^2(x)sec2(x) is tan⁡(x)\tan(x)tan(x).So, we have:∫6 dx=6x\int 6 \, dx = 6x∫6dx=6x ∫sec⁡2(x) dx=tan⁡(x)\int \sec^2(x) \, dx = \tan(x)∫sec2(x)dx=tan(x)

STEP 5

Combine the results and add the constant of integration C C C:∫(6+sec⁡2(x)) dx=6x+tan⁡(x)+C\int (6 + \sec^2(x)) \, dx = 6x + \tan(x) + C∫(6+sec2(x))dx=6x+tan(x)+C

STEP 6

STEP 7

STEP 8

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QuestionFind the following antiderivatives:
1. $\int 6+\sec ^{2}(x) d x=\square+C$ .
2. $\int \frac{7 x^{2}-6}{x^{2}} d x=\square+C$ .

1. We are finding indefinite integrals, which include a constant of integration $C$ .
2. We will use basic integration rules and properties to solve the integrals.

1. Solve the first integral: $\int (6 + \sec^2(x)) \, dx$ .
2. Solve the second integral: $\int \frac{7x^2 - 6}{x^2} \, dx$ .

Integrate each term separately:
1. The integral of a constant $6$ is $6x$ .
2. The integral of $\sec^2(x)$ is $\tan(x)$ .
So, we have:
$\int 6 \, dx = 6x$ $\int \sec^2(x) \, dx = \tan(x)$

Combine the results and add the constant of integration $C$ :
$\int (6 + \sec^2(x)) \, dx = 6x + \tan(x) + C$