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To find the integral of the expression [tex]\( x^5 \sqrt{x^3 + 9} \, dx \)[/tex], we need to proceed methodically. Here is a detailed step-by-step solution:
1. Simplify the Integrand:
We'll first rewrite the integrand [tex]\( x^5 \sqrt{x^3 + 9} \)[/tex] for better clarity.
[tex]\[ x^5 \sqrt{x^3 + 9} \][/tex]
2. Substitute to Simplify:
Let's consider a substitution [tex]\( u = x^3 + 9 \)[/tex]. Therefore, [tex]\( \frac{du}{dx} = 3x^2 \)[/tex] or [tex]\( du = 3x^2 \, dx \)[/tex]. Solving for [tex]\( dx \)[/tex], we get:
[tex]\[ dx = \frac{du}{3x^2} \][/tex]
3. Express [tex]\( x \)[/tex] in Terms of [tex]\( u \)[/tex]:
From [tex]\( u = x^3 + 9 \)[/tex], we have [tex]\( x^3 = u - 9 \)[/tex]. Hence,
[tex]\[ x^6 = (x^3)^2 = (u - 9)^2 \][/tex]
4. Rewrite the Integral:
Substitute [tex]\( u \)[/tex] and [tex]\( dx \)[/tex] into the integral:
[tex]\[ \int x^5 \sqrt{x^3 + 9} \, dx = \int x^5 \sqrt{u} \cdot \frac{du}{3x^2} \][/tex]
Simplify the integrand:
[tex]\[ \int x^5 \sqrt{u} \cdot \frac{du}{3x^2} = \int \frac{x^5 \sqrt{u}}{3x^2} \, du = \int \frac{x^3 x^2 \sqrt{u}}{3x^2} \, du = \int \frac{x^3 \sqrt{u}}{3} \, du \][/tex]
As [tex]\( x^3 = u - 9 \)[/tex], the integrand becomes:
[tex]\[ \int \frac{(u - 9) \sqrt{u}}{3} \, du \][/tex]
5. Distribute and Simplify:
Distribute [tex]\( \sqrt{u} \)[/tex] and split the integral:
[tex]\[ \int \frac{(u - 9) \sqrt{u}}{3} \, du = \int \frac{u^{3/2} - 9u^{1/2}}{3} \, du = \frac{1}{3} \int (u^{3/2} - 9u^{1/2}) \, du \][/tex]
Separate the integrals:
[tex]\[ \frac{1}{3} \left( \int u^{3/2} \, du - 9 \int u^{1/2} \, du \right) \][/tex]
6. Evaluate the Integrals:
Integrate each term separately:
[tex]\[ \int u^{3/2} \, du = \frac{2}{5} u^{5/2} \][/tex]
[tex]\[ \int u^{1/2} \, du = \frac{2}{3} u^{3/2} \][/tex]
Substitute back:
[tex]\[ \frac{1}{3} \left( \frac{2}{5} u^{5/2} - 9 \cdot \frac{2}{3} u^{3/2} \right) \][/tex]
Simplify:
[tex]\[ \frac{1}{3} \left( \frac{2}{5} u^{5/2} - 6 u^{3/2} \right) \][/tex]
[tex]\[ \frac{1}{3} \cdot \frac{2}{5} u^{5/2} - \frac{1}{3} \cdot 6 u^{3/2} = \frac{2}{15} u^{5/2} - 2 u^{3/2} \][/tex]
7. Substitute [tex]\( u \)[/tex] Back to [tex]\( x \)[/tex]:
Recall [tex]\( u = x^3 + 9 \)[/tex]:
[tex]\[ \frac{2}{15} (x^3 + 9)^{5/2} - 2 (x^3 + 9)^{3/2} \][/tex]
Putting everything together, the integral is:
[tex]\[ \int x^5 \sqrt{x^3 + 9} \, dx = \frac{2}{15} x^6 \sqrt{x^3 + 9} + \frac{2}{5} x^3 \sqrt{x^3 + 9} - \frac{36}{5} \sqrt{x^3 + 9} + C \][/tex]
Where [tex]\( C \)[/tex] is the constant of integration.
1. Simplify the Integrand:
We'll first rewrite the integrand [tex]\( x^5 \sqrt{x^3 + 9} \)[/tex] for better clarity.
[tex]\[ x^5 \sqrt{x^3 + 9} \][/tex]
2. Substitute to Simplify:
Let's consider a substitution [tex]\( u = x^3 + 9 \)[/tex]. Therefore, [tex]\( \frac{du}{dx} = 3x^2 \)[/tex] or [tex]\( du = 3x^2 \, dx \)[/tex]. Solving for [tex]\( dx \)[/tex], we get:
[tex]\[ dx = \frac{du}{3x^2} \][/tex]
3. Express [tex]\( x \)[/tex] in Terms of [tex]\( u \)[/tex]:
From [tex]\( u = x^3 + 9 \)[/tex], we have [tex]\( x^3 = u - 9 \)[/tex]. Hence,
[tex]\[ x^6 = (x^3)^2 = (u - 9)^2 \][/tex]
4. Rewrite the Integral:
Substitute [tex]\( u \)[/tex] and [tex]\( dx \)[/tex] into the integral:
[tex]\[ \int x^5 \sqrt{x^3 + 9} \, dx = \int x^5 \sqrt{u} \cdot \frac{du}{3x^2} \][/tex]
Simplify the integrand:
[tex]\[ \int x^5 \sqrt{u} \cdot \frac{du}{3x^2} = \int \frac{x^5 \sqrt{u}}{3x^2} \, du = \int \frac{x^3 x^2 \sqrt{u}}{3x^2} \, du = \int \frac{x^3 \sqrt{u}}{3} \, du \][/tex]
As [tex]\( x^3 = u - 9 \)[/tex], the integrand becomes:
[tex]\[ \int \frac{(u - 9) \sqrt{u}}{3} \, du \][/tex]
5. Distribute and Simplify:
Distribute [tex]\( \sqrt{u} \)[/tex] and split the integral:
[tex]\[ \int \frac{(u - 9) \sqrt{u}}{3} \, du = \int \frac{u^{3/2} - 9u^{1/2}}{3} \, du = \frac{1}{3} \int (u^{3/2} - 9u^{1/2}) \, du \][/tex]
Separate the integrals:
[tex]\[ \frac{1}{3} \left( \int u^{3/2} \, du - 9 \int u^{1/2} \, du \right) \][/tex]
6. Evaluate the Integrals:
Integrate each term separately:
[tex]\[ \int u^{3/2} \, du = \frac{2}{5} u^{5/2} \][/tex]
[tex]\[ \int u^{1/2} \, du = \frac{2}{3} u^{3/2} \][/tex]
Substitute back:
[tex]\[ \frac{1}{3} \left( \frac{2}{5} u^{5/2} - 9 \cdot \frac{2}{3} u^{3/2} \right) \][/tex]
Simplify:
[tex]\[ \frac{1}{3} \left( \frac{2}{5} u^{5/2} - 6 u^{3/2} \right) \][/tex]
[tex]\[ \frac{1}{3} \cdot \frac{2}{5} u^{5/2} - \frac{1}{3} \cdot 6 u^{3/2} = \frac{2}{15} u^{5/2} - 2 u^{3/2} \][/tex]
7. Substitute [tex]\( u \)[/tex] Back to [tex]\( x \)[/tex]:
Recall [tex]\( u = x^3 + 9 \)[/tex]:
[tex]\[ \frac{2}{15} (x^3 + 9)^{5/2} - 2 (x^3 + 9)^{3/2} \][/tex]
Putting everything together, the integral is:
[tex]\[ \int x^5 \sqrt{x^3 + 9} \, dx = \frac{2}{15} x^6 \sqrt{x^3 + 9} + \frac{2}{5} x^3 \sqrt{x^3 + 9} - \frac{36}{5} \sqrt{x^3 + 9} + C \][/tex]
Where [tex]\( C \)[/tex] is the constant of integration.
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