Of course! Let's solve the integral [tex]\(\int y^2 \sqrt{y} \, dy\)[/tex] step by step.
1. Combine the terms inside the integral: We can start by rewriting the integrand [tex]\(y^2 \sqrt{y}\)[/tex] in a simpler form. Recall that [tex]\(\sqrt{y}\)[/tex] is the same as [tex]\(y^{1/2}\)[/tex]. Therefore,
[tex]\[
y^2 \sqrt{y} = y^2 \cdot y^{1/2} = y^{2 + 1/2} = y^{5/2}
\][/tex]
2. Set up the integral: Now that we have simplified the expression, the integral becomes
[tex]\[
\int y^{5/2} \, dy
\][/tex]
3. Integrate using the power rule: The power rule for integration states that [tex]\(\int y^n \, dy = \frac{y^{n+1}}{n+1} + C\)[/tex], where [tex]\(n \neq -1\)[/tex]. In our case, [tex]\(n = \frac{5}{2}\)[/tex]:
[tex]\[
\int y^{5/2} \, dy = \frac{y^{(5/2) + 1}}{(5/2) + 1} + C
\][/tex]
4. Simplify the exponent and the fraction: Calculate the new exponent and simplify the fraction in the denominator:
[tex]\[
(5/2) + 1 = \frac{5}{2} + \frac{2}{2} = \frac{7}{2}
\][/tex]
So the integral becomes:
[tex]\[
\frac{y^{7/2}}{7/2} + C
\][/tex]
5. Simplify the fraction: Dividing by [tex]\(\frac{7}{2}\)[/tex] is the same as multiplying by [tex]\(\frac{2}{7}\)[/tex]:
[tex]\[
\frac{y^{7/2}}{7/2} = y^{7/2} \cdot \frac{2}{7} = \frac{2y^{7/2}}{7}
\][/tex]
6. Write the final answer: Combine everything to get the final result:
[tex]\[
\int y^2 \sqrt{y} \, dy = \frac{2y^{7/2}}{7} + C
\][/tex]
Thus, the solution to the integral [tex]\(\int y^2 \sqrt{y} \, dy\)[/tex] is [tex]\(\frac{2y^{7/2}}{7} + C\)[/tex].
