Problem 52
Question
Evaluate the integrals. $$ \int_{0}^{1} e^{1+x^{2}} x d x $$
Step-by-Step Solution
Verified Answer
\( \frac{e^2 - e}{2} \)
1Step 1: Identify the Integral Structure
The integral we need to solve is \( \int_{0}^{1} e^{1+x^{2}} x \, dx \). Notice that the integrand is \( e^{1+x^{2}} x \). This suggests trying substitution because of the term \( x \), which often pairs well with derivatives.
2Step 2: Choose a Suitable Substitution
Consider the substitution \( u = 1 + x^2 \). Then, the differential \( du = 2x \, dx \) implies \( x \, dx = \frac{1}{2} du \). The limits also change: when \( x = 0 \), \( u = 1 \); and when \( x = 1 \), \( u = 2 \).
3Step 3: Substitute and Rewrite the Integral
Using the substitution, rewrite the integral. The original integral becomes: \[\int_{1}^{2} e^{u} \frac{1}{2} \, du\] Simplifying, we have: \[\frac{1}{2} \int_{1}^{2} e^{u} \, du\]
4Step 4: Evaluate the New Integral
Evaluate \( \frac{1}{2} \int_{1}^{2} e^{u} \, du \). The antiderivative of \( e^u \) is simply \( e^u \). So, we have: \[\frac{1}{2} \left[ e^u \right]_{1}^{2} = \frac{1}{2} (e^2 - e^1)\]
5Step 5: Simplify the Result
Finally, simplify the result obtained after evaluating the limits: \[\frac{1}{2} (e^2 - e) = \frac{e^2 - e}{2}\]
Key Concepts
definite integralu-substitutionantiderivativelimits of integration
definite integral
A definite integral is not just about finding the area under a curve; it is about calculating a specific value that represents this area over a certain interval. The notation \( \int_{a}^{b} f(x) \; dx \) indicates a definite integral, with \( a \) and \( b \) as limits of integration. These limits specify the start and end points on the \( x \)-axis.
In the given exercise, the definite integral \( \int_{0}^{1} e^{1+x^{2}} x \, dx \) reflects the area under the curve of \( e^{1+x^2} x \) from \( x=0 \) to \( x=1 \). By calculating this value, you not only understand the behavior of the function over that interval but also its total accumulation.
In the given exercise, the definite integral \( \int_{0}^{1} e^{1+x^{2}} x \, dx \) reflects the area under the curve of \( e^{1+x^2} x \) from \( x=0 \) to \( x=1 \). By calculating this value, you not only understand the behavior of the function over that interval but also its total accumulation.
- The definite integral results in a real number.
- It requires finding the antiderivative first, followed by evaluating it at the upper and lower limits.
- A change in limits due to substitution is common during evaluation.
u-substitution
U-substitution is a technique used to simplify the calculation of integrals, similar to the chain rule in differentiation but in reverse. This method involves substituting a part of the integrand with a new variable \( u \), transforming the integral into a simpler form.
For the integral \( \int_{0}^{1} e^{1+x^{2}} x \, dx \), the substitution \( u = 1 + x^2 \) simplifies the integral as it turns the complex exponent into a simple linear term \( u \). With this substitution:
For the integral \( \int_{0}^{1} e^{1+x^{2}} x \, dx \), the substitution \( u = 1 + x^2 \) simplifies the integral as it turns the complex exponent into a simple linear term \( u \). With this substitution:
- \( du = 2x \, dx \), leading to \( x \, dx = \frac{1}{2} \, du \).
- The limits of integration change correspondingly \( (x=0 \to u=1, \; x=1 \to u=2) \).
antiderivative
An antiderivative is essentially the reverse process of taking a derivative. It is crucial in calculating definite integrals since it forms the basis of the Fundamental Theorem of Calculus.
To solve \( \frac{1}{2} \int_{1}^{2} e^{u} \, du \) in the original problem, we directly find the antiderivative of \( e^u \), which is simply \( e^u \) itself. It means you are looking for a function whose derivative matches the integrand. Here, since \( e^u \) differentiates to itself, it acts as its own antiderivative.
To solve \( \frac{1}{2} \int_{1}^{2} e^{u} \, du \) in the original problem, we directly find the antiderivative of \( e^u \), which is simply \( e^u \) itself. It means you are looking for a function whose derivative matches the integrand. Here, since \( e^u \) differentiates to itself, it acts as its own antiderivative.
- Use the antiderivative in evaluating the definite integral over the new limits of integration \( \left[ e^u \right]_{1}^{2} \).
- Antiderivatives restore the function to its prior accumulated state, leading to evaluating it within specific bounds.
limits of integration
Limits of integration are central in defining the scope of the definite integral. They pinpoint the interval on the \( x \)-axis over which you measure the area.
In the original exercise, limits changed due to the \( u \)-substitution process:
Properly understanding and handling limits help ensure accurate calculations as they adjust the integral to the new scope provided by substitution.
In the original exercise, limits changed due to the \( u \)-substitution process:
- Initially, the limits were \( x=0 \) and \( x=1 \).
- After substitution, they transformed into \( u=1 \) and \( u=2 \).
Properly understanding and handling limits help ensure accurate calculations as they adjust the integral to the new scope provided by substitution.
Other exercises in this chapter
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