Math / Calculus

QuestionDetermine whether the following statements are true and give an explanation or counterexample. Assume that $f, f^{\prime}$ , and $f^{\prime \prime}$ and are continuous functions for all real numbers. Complete parts (a) through (e) below. a. Decide whether the statement $\int f(x) f^{\prime}(x) d x=\frac{1}{2}(f(x))^{2}+C$ is correct. Choose the correct answer below. A. True; $\int f(x) f^{\prime}(x) d x=\frac{1}{2}(f(x))^{2}+C$ B. False; $\int f(x) f^{\prime}(x) d x=2(f(x))^{2}+C$ C. False; $\int f(x) f^{\prime}(x) d x=\frac{1}{2}(f(x))^{2}\left(f^{\prime \prime}(x)\right)^{2}+C$ D. False; $\int f(x) f^{\prime}(x) d x=\frac{1}{2}(f(x))+C$

Studdy Solution

STEP 1

1. $f, f^{\prime},$ and $f^{\prime \prime}$ are continuous functions for all real numbers.
2. We need to evaluate the correctness of the integral expression $\int f(x) f^{\prime}(x) \, dx$ .

STEP 2

1. Identify the integration technique applicable to the given integral.
2. Apply the technique to evaluate the integral.
3. Compare the result with the given options.

STEP 3

Recognize that the integral $\int f(x) f^{\prime}(x) \, dx$ can be solved using the method of integration by substitution.

STEP 4

Let $u = f(x)$ , then $du = f^{\prime}(x) \, dx$ .
The integral becomes $\int u \, du$ .

STEP 5

Evaluate the integral $\int u \, du$ .
The result is $\frac{1}{2} u^2 + C$ .

STEP 6

Substitute back $u = f(x)$ into the result.
The integral evaluates to $\frac{1}{2} (f(x))^2 + C$ .

STEP 7

Compare the evaluated result $\frac{1}{2} (f(x))^2 + C$ with the given options.
Option A matches the evaluated result.
The statement is:
A. True; $\int f(x) f^{\prime}(x) \, dx = \frac{1}{2}(f(x))^{2} + C$

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Studdy Solution

STEP 1

1. $f, f^{\prime},$ and $f^{\prime \prime}$ are continuous functions for all real numbers.
2. We need to evaluate the correctness of the integral expression $\int f(x) f^{\prime}(x) \, dx$ .

STEP 2

1. Identify the integration technique applicable to the given integral.
2. Apply the technique to evaluate the integral.
3. Compare the result with the given options.

STEP 3

Recognize that the integral $\int f(x) f^{\prime}(x) \, dx$ can be solved using the method of integration by substitution.

STEP 4

Let $u = f(x)$ , then $du = f^{\prime}(x) \, dx$ .
The integral becomes $\int u \, du$ .

STEP 5

Evaluate the integral $\int u \, du$ .
The result is $\frac{1}{2} u^2 + C$ .

STEP 6

Substitute back $u = f(x)$ into the result.
The integral evaluates to $\frac{1}{2} (f(x))^2 + C$ .

STEP 7

Compare the evaluated result $\frac{1}{2} (f(x))^2 + C$ with the given options.
Option A matches the evaluated result.
The statement is:
A. True; $\int f(x) f^{\prime}(x) \, dx = \frac{1}{2}(f(x))^{2} + C$

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Studdy Solution

STEP 1

1. f,f′, f, f^{\prime}, f,f′, and f′′ f^{\prime \prime} f′′ are continuous functions for all real numbers.2. We need to evaluate the correctness of the integral expression ∫f(x)f′(x) dx \int f(x) f^{\prime}(x) \, dx ∫f(x)f′(x)dx.

STEP 2

1. Identify the integration technique applicable to the given integral.2. Apply the technique to evaluate the integral.3. Compare the result with the given options.

STEP 3

Recognize that the integral ∫f(x)f′(x) dx \int f(x) f^{\prime}(x) \, dx ∫f(x)f′(x)dx can be solved using the method of integration by substitution.

STEP 4

Let u=f(x) u = f(x) u=f(x), then du=f′(x) dx du = f^{\prime}(x) \, dx du=f′(x)dx.The integral becomes ∫u du \int u \, du ∫udu.

STEP 5

Evaluate the integral ∫u du \int u \, du ∫udu.The result is 12u2+C \frac{1}{2} u^2 + C 21​u2+C.

STEP 6

Substitute back u=f(x) u = f(x) u=f(x) into the result.The integral evaluates to 12(f(x))2+C \frac{1}{2} (f(x))^2 + C 21​(f(x))2+C.

STEP 7

Compare the evaluated result 12(f(x))2+C \frac{1}{2} (f(x))^2 + C 21​(f(x))2+C with the given options.Option A matches the evaluated result. The statement is:A. True; ∫f(x)f′(x) dx=12(f(x))2+C\int f(x) f^{\prime}(x) \, dx = \frac{1}{2}(f(x))^{2} + C∫f(x)f′(x)dx=21​(f(x))2+C

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1. $f, f^{\prime},$ and $f^{\prime \prime}$ are continuous functions for all real numbers.
2. We need to evaluate the correctness of the integral expression $\int f(x) f^{\prime}(x) \, dx$ .

1. Identify the integration technique applicable to the given integral.
2. Apply the technique to evaluate the integral.
3. Compare the result with the given options.

Recognize that the integral $\int f(x) f^{\prime}(x) \, dx$ can be solved using the method of integration by substitution.

Let $u = f(x)$ , then $du = f^{\prime}(x) \, dx$ .
The integral becomes $\int u \, du$ .

Evaluate the integral $\int u \, du$ .
The result is $\frac{1}{2} u^2 + C$ .

Substitute back $u = f(x)$ into the result.
The integral evaluates to $\frac{1}{2} (f(x))^2 + C$ .

Compare the evaluated result $\frac{1}{2} (f(x))^2 + C$ with the given options.
Option A matches the evaluated result.
The statement is:
A. True; $\int f(x) f^{\prime}(x) \, dx = \frac{1}{2}(f(x))^{2} + C$