Math Problem Statement

Evaluate R 2 0 xex 2 dx.

Solution

To evaluate the integral R=02xex2dxR = \int_{0}^{2} x e^{x^2} \, dx, we can use the technique of substitution.

Step 1: Substitution

Let u=x2u = x^2. Then, the derivative of uu with respect to xx is: dudx=2x\frac{du}{dx} = 2x So, du=2xdxdu = 2x \, dx, or 12du=xdx\frac{1}{2} du = x \, dx.

Step 2: Rewrite the Integral

Substitute into the integral: R=02xex2dx=u(0)u(2)eu12duR = \int_{0}^{2} x e^{x^2} \, dx = \int_{u(0)}^{u(2)} e^{u} \cdot \frac{1}{2} du Given u=x2u = x^2, when x=0x = 0, u=0u = 0, and when x=2x = 2, u=4u = 4. Therefore, the integral becomes: R=1204euduR = \frac{1}{2} \int_{0}^{4} e^u \, du

Step 3: Evaluate the Integral

The integral of eue^u with respect to uu is simply eue^u. So, we have: R=12[eu]04=12(e4e0)R = \frac{1}{2} \left[ e^u \right]_{0}^{4} = \frac{1}{2} \left( e^4 - e^0 \right)

Step 4: Simplify the Result

Since e0=1e^0 = 1, the final expression is: R=12(e41)R = \frac{1}{2} \left( e^4 - 1 \right)

So, the evaluated integral is: R=e412R = \frac{e^4 - 1}{2}

Would you like more details or have any questions?

Related Questions:

  1. How would the integral change if the limits were different, say from 1 to 3?
  2. Can the method of integration by parts be applied to this integral?
  3. How does substitution work in integrals involving trigonometric functions?
  4. What is the significance of the constant factor in definite integrals?
  5. How does the exponential function behave under integration compared to polynomial functions?

Tip: Substitution is a powerful technique for solving integrals where the integrand is a composite function, especially when one part of the function is the derivative of another.

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Math Problem Analysis

Mathematical Concepts

Integration
Substitution
Exponential Functions

Formulas

Substitution in integrals

Theorems

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Suitable Grade Level

Advanced High School to College