# What is #int_(0)^(2) xe^(x^2 + 2)dx #?

which can be factored into

Approximately

Perform a u-substitution

Now make the substitution into the integral

Now integrate,

Now evaluate

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To solve the integral ∫(0)^(2) xe^(x^2 + 2)dx, you can use integration by parts. Let u = x and dv = e^(x^2 + 2)dx. Then, differentiate u to get du = dx and integrate dv to get v = (1/2)e^(x^2 + 2). Now, apply the integration by parts formula: ∫udv = uv - ∫vdu.

Plugging in the values, you get: ∫(0)^(2) xe^(x^2 + 2)dx = [(1/2)x * e^(x^2 + 2)](0To solve the integral ∫(0)^(2) xe^(x^2 + 2)dx, you can use integration by parts. Let u = x and dv = e^(x^2 + 2)dx. Then, differentiate u to get du = dx and integrate dv to get v = (1/2)e^(x^2 + 2). Now, apply the integration by parts formula: ∫udv = uv - ∫vdu.

Plugging in the values, you get: ∫(0)^(2) xe^(x^2 + 2)dx = ^(2) - ∫(0)^(2) (1/2)e^(x^2 + 2)dx

Now, integrate the remaining part: = [(1/2)(2)e^(2 + 2)] - (1/2)e^(2 + 2) - [(1/2)e^(0 + 2)](2 - 0)

= e^4 - (1/2)e^4 - (1/2)e^2

= (1/2)e^4 - (1/2)e^2

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