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Magazine Chapter 7: Problem 35
Evaluate the integrals. $$\int_{0}^{3}(\sqrt{2}+1) x^{\sqrt{2}} d x$$
Short Answer
Expert verified
The integral evaluates to approximately 12.18034.
Step by step solution
01
Identify the Integral Type
The given integral is \( \int_{0}^{3} (\sqrt{2} + 1) x^{\sqrt{2}} \, dx \). This is a definite integral involving a power of \( x \). We'll use the power rule for integration.
02
Apply the Power Rule for Integration
The power rule states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for any real number \( n eq -1 \). Here, \( n = \sqrt{2} \), so integrating \( x^{\sqrt{2}} \) gives us \( \frac{x^{\sqrt{2} + 1}}{\sqrt{2} + 1} \). Don't forget to include the constant factor \( (\sqrt{2} + 1) \) from the original integral.
03
Integrate the Function
Integrating \( (\sqrt{2} + 1)x^{\sqrt{2}} \) over \( x \) gives:\[(\sqrt{2} + 1) \cdot \frac{x^{\sqrt{2} + 1}}{\sqrt{2} + 1} = x^{\sqrt{2} + 1}\]
04
Evaluate the Definite Integral
We evaluate the antiderivative at the upper and lower limits of the integral. This gives us:\[ \left[ x^{\sqrt{2} + 1} \right]_{0}^{3} = 3^{\sqrt{2} + 1} - 0^{\sqrt{2} + 1} \] Since \( 0^{\sqrt{2} + 1} = 0 \), we only need the value at 3.
05
Calculate the Final Answer
Finally, calculate:\[ 3^{\sqrt{2} + 1} \approx 12.18034 \] Thus, the integral evaluates approximately to 12.18034.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Definite Integral
A definite integral represents the area under a curve within a set range, known as the bounds. For the integral \( \int_{a}^{b} f(x) \, dx \), these bounds are \( a \) and \( b \), meaning we evaluate the function starting at \( x = a \) and ending at \( x = b \). This is different from an indefinite integral, where we find a general expression of the antiderivative without any bounds.
- In a definite integral, the limits provide numerical values that make the result specific rather than general.
- Definite integrals are used to calculate total summations, such as area, volume, and other accumulative values.
- The result of a definite integral is a number, not a function.
Power Rule for Integration
The power rule is a fundamental rule in calculus used to integrate functions of the form \( x^n \). It states that:\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \text{ for any real number } n eq -1.\]This rule allows us to easily find the antiderivative of any power function. In the given problem, the function \( x^{\sqrt{2}} \) is integrated by applying:
- Add 1 to the exponent \( n \).
- Divide by the new exponent \( n+1 \).
- Remember the constant \( C \) is omitted within definite integrals.
Antiderivative
An antiderivative is essentially the reverse process of differentiation. If you have a function, finding its antiderivative means figuring out what original function it came from, such that if you took its derivative, you would end up back with the initial function.
- The antiderivative of \( f(x) \) when integrated results in a function \( F(x) \), so that \( F'(x) = f(x) \).
- When dealing with definite integrals, you consider only the definite part, so the additive constant \( C \) gets cancelled out.
Evaluating Integrals
Once the function has been integrated and its antiderivative is known, the next step in solving a definite integral is to evaluate it. This means taking the antiderivative and substituting the upper bound and lower bound values into it.
- For a definite integral \( \int_{a}^{b} f(x) \, dx \), evaluate as \( F(b) - F(a) \), where \( F(x) \) is the antiderivative of \( f(x) \).
- This calculation results in a single, precise numerical value.
