Math / Calculus

QuestionEvaluate the integral. HINT [See Example 2.] $\int_{0}^{1} 5 e^{x} d x$ $\square$

Studdy Solution

STEP 1

1. The integral to evaluate is a definite integral from 0 to 1.
2. The integrand is $5e^x$ , which is a standard exponential function.
3. We will use the fundamental theorem of calculus to evaluate the integral.

STEP 2

1. Identify the antiderivative of the integrand.
2. Evaluate the antiderivative at the upper and lower limits.
3. Subtract the values obtained in step 2 to find the value of the definite integral.

STEP 3

First, identify the antiderivative of the integrand $5e^x$ . The antiderivative of $e^x$ is $e^x$ , so the antiderivative of $5e^x$ is:
$\int 5e^x \, dx = 5e^x + C$
where $C$ is the constant of integration.

STEP 4

Now, evaluate the antiderivative at the upper limit $x = 1$ :
$5e^1 = 5e$
Evaluate the antiderivative at the lower limit $x = 0$ :
$5e^0 = 5 \cdot 1 = 5$

STEP 5

Subtract the value of the antiderivative at the lower limit from the value at the upper limit:
$5e - 5$
Thus, the value of the definite integral is:
$\boxed{5(e - 1)}$
The value of the integral is:
$\boxed{5(e - 1)}$

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QuestionEvaluate the integral. HINT [See Example 2.] $\int_{0}^{1} 5 e^{x} d x$ $\square$

Studdy Solution

STEP 1

1. The integral to evaluate is a definite integral from 0 to 1.
2. The integrand is $5e^x$ , which is a standard exponential function.
3. We will use the fundamental theorem of calculus to evaluate the integral.

STEP 2

1. Identify the antiderivative of the integrand.
2. Evaluate the antiderivative at the upper and lower limits.
3. Subtract the values obtained in step 2 to find the value of the definite integral.

STEP 3

First, identify the antiderivative of the integrand $5e^x$ . The antiderivative of $e^x$ is $e^x$ , so the antiderivative of $5e^x$ is:
$\int 5e^x \, dx = 5e^x + C$
where $C$ is the constant of integration.

STEP 4

Now, evaluate the antiderivative at the upper limit $x = 1$ :
$5e^1 = 5e$
Evaluate the antiderivative at the lower limit $x = 0$ :
$5e^0 = 5 \cdot 1 = 5$

STEP 5

Subtract the value of the antiderivative at the lower limit from the value at the upper limit:
$5e - 5$
Thus, the value of the definite integral is:
$\boxed{5(e - 1)}$
The value of the integral is:
$\boxed{5(e - 1)}$

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QuestionEvaluate the integral. HINT [See Example 2.] ∫015exdx\int_{0}^{1} 5 e^{x} d x∫01​5exdx □\square□

Studdy Solution

STEP 1

1. The integral to evaluate is a definite integral from 0 to 1.2. The integrand is 5ex 5e^x 5ex, which is a standard exponential function.3. We will use the fundamental theorem of calculus to evaluate the integral.

STEP 2

1. Identify the antiderivative of the integrand.2. Evaluate the antiderivative at the upper and lower limits.3. Subtract the values obtained in step 2 to find the value of the definite integral.

STEP 3

First, identify the antiderivative of the integrand 5ex 5e^x 5ex. The antiderivative of ex e^x ex is ex e^x ex, so the antiderivative of 5ex 5e^x 5ex is:∫5ex dx=5ex+C \int 5e^x \, dx = 5e^x + C ∫5exdx=5ex+Cwhere C C C is the constant of integration.

STEP 4

Now, evaluate the antiderivative at the upper limit x=1 x = 1 x=1:5e1=5e 5e^1 = 5e 5e1=5eEvaluate the antiderivative at the lower limit x=0 x = 0 x=0:5e0=5⋅1=5 5e^0 = 5 \cdot 1 = 5 5e0=5⋅1=5

STEP 5

Subtract the value of the antiderivative at the lower limit from the value at the upper limit:5e−5 5e - 5 5e−5Thus, the value of the definite integral is:5(e−1) \boxed{5(e - 1)} 5(e−1)​ The value of the integral is:5(e−1) \boxed{5(e - 1)} 5(e−1)​

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QuestionEvaluate the integral. HINT [See Example 2.] $\int_{0}^{1} 5 e^{x} d x$ $\square$

1. The integral to evaluate is a definite integral from 0 to 1.
2. The integrand is $5e^x$ , which is a standard exponential function.
3. We will use the fundamental theorem of calculus to evaluate the integral.

1. Identify the antiderivative of the integrand.
2. Evaluate the antiderivative at the upper and lower limits.
3. Subtract the values obtained in step 2 to find the value of the definite integral.

First, identify the antiderivative of the integrand $5e^x$ . The antiderivative of $e^x$ is $e^x$ , so the antiderivative of $5e^x$ is:
$\int 5e^x \, dx = 5e^x + C$
where $C$ is the constant of integration.

Now, evaluate the antiderivative at the upper limit $x = 1$ :
$5e^1 = 5e$
Evaluate the antiderivative at the lower limit $x = 0$ :
$5e^0 = 5 \cdot 1 = 5$

Subtract the value of the antiderivative at the lower limit from the value at the upper limit:
$5e - 5$
Thus, the value of the definite integral is:
$\boxed{5(e - 1)}$
The value of the integral is:
$\boxed{5(e - 1)}$