Question

observe that the given integrand, $x^{4}+6y^{5}$ is a function in two variables. consider $x$ as a constant. integrate with respect to $x^{4}+6y^{5}$ using the power rule of integration. $\int_{\sqrt{x}}^{x^{5}}(x^{4}+6y^{5})dy = \sqrt{x}_{\sqrt{x}}^{x^{5}}$

Explanation:

Step1: Integrate with respect to y

Treat \( x \) as a constant. The integral of \( x^4 \) with respect to \( y \) is \( x^4 y \), and the integral of \( 6y^5 \) with respect to \( y \) is \( 6 \cdot \frac{y^6}{6}=y^6 \). So the antiderivative is \( x^4 y + y^6 \).

Step2: Evaluate the definite integral

Evaluate \( x^4 y + y^6 \) from \( y = \sqrt{x} \) to \( y = x^5 \).
First, substitute \( y = x^5 \): \( x^4 \cdot x^5 + (x^5)^6 = x^9 + x^{30} \).
Then, substitute \( y = \sqrt{x} \): \( x^4 \cdot \sqrt{x} + (\sqrt{x})^6 = x^4 \cdot x^{\frac{1}{2}} + x^3 = x^{\frac{9}{2}} + x^3 \).
Subtract the lower limit from the upper limit: \( (x^9 + x^{30}) - (x^{\frac{9}{2}} + x^3) = x^9 + x^{30} - x^{\frac{9}{2}} - x^3 \).

Answer:

\( x^9 + x^{30} - x^{\frac{9}{2}} - x^3 \)