Math  /  Calculus

QuestionEvaluate. (Be sure to check by differentiating!) x4ex5dx\int x^{4} e^{x^{5}} d x
Determine a change of variables from x to u . Choose the correct answer below. A. u=exu=e^{\mathrm{x}} B. u=x4exu=x^{4} e^{x} C. u=x5u=x^{5} D. u=x4u=x^{4}

Studdy Solution
Differentiate the result to verify the integration:
Differentiate 15ex5+C \frac{1}{5} e^{x^{5}} + C with respect to x x :
ddx(15ex5+C)=15ex5ddx(x5) \frac{d}{dx} \left( \frac{1}{5} e^{x^{5}} + C \right) = \frac{1}{5} \cdot e^{x^{5}} \cdot \frac{d}{dx}(x^{5})
=15ex55x4 = \frac{1}{5} \cdot e^{x^{5}} \cdot 5x^{4}
=x4ex5 = x^{4} e^{x^{5}}
This matches the original integrand, confirming the integration is correct.
The correct substitution is:
u=x5 \boxed{u = x^{5}}

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