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Sagot :
To evaluate the definite integral
[tex]\[ \int_1^{49} \frac{(3 + \sqrt{x})^{4 / 3}}{\sqrt{x}} \, dx, \][/tex]
we follow these steps:
1. Substitution:
Let [tex]\( u = \sqrt{x} \)[/tex]. Therefore, [tex]\( u^2 = x \)[/tex] and hence [tex]\( du = \frac{1}{2\sqrt{x}} dx \)[/tex]. This implies [tex]\( dx = 2u \, du \)[/tex].
2. Adjusting Limits of Integration:
When [tex]\( x = 1 \)[/tex], [tex]\( u = \sqrt{1} = 1 \)[/tex].
When [tex]\( x = 49 \)[/tex], [tex]\( u = \sqrt{49} = 7 \)[/tex].
3. Rewrite the Integrand:
Substituting [tex]\( x = u^2 \)[/tex] and [tex]\( dx = 2u \, du \)[/tex], the integrand becomes:
[tex]\[ \frac{(3 + \sqrt{x})^{4 / 3}}{\sqrt{x}} \cdot dx = \frac{(3 + u)^{4 / 3}}{u} \cdot 2u \, du = 2(3 + u)^{4 / 3} \, du \][/tex]
4. New Integral in Terms of [tex]\( u \)[/tex]:
The integral is now:
[tex]\[ \int_1^{49} \frac{(3 + \sqrt{x})^{4 / 3}}{\sqrt{x}} \, dx = \int_1^7 2 (3 + u)^{4 / 3} \, du \][/tex]
5. Simplify and Evaluate:
Factor out the constant:
[tex]\[ 2 \int_1^7 (3 + u)^{4 / 3} \, du \][/tex]
Let [tex]\( v = 3 + u \)[/tex]. Thus, [tex]\( du = dv \)[/tex].
When [tex]\( u = 1 \)[/tex], [tex]\( v = 3 + 1 = 4 \)[/tex].
When [tex]\( u = 7 \)[/tex], [tex]\( v = 3 + 7 = 10 \)[/tex].
Substituting the limits and the integrand:
[tex]\[ 2 \int_4^{10} v^{4/3} \, dv \][/tex]
6. Integrate:
Use the power rule for integration:
[tex]\[ \int v^{4/3} \, dv = \frac{v^{(4/3)+1}}{(4/3)+1} = \frac{v^{7/3}}{7/3} = \frac{3}{7} v^{7/3} \][/tex]
So, the integral becomes:
[tex]\[ 2 \left[ \frac{3}{7} v^{7/3} \right]_4^{10} \][/tex]
7. Evaluate the Definite Integral:
Substitute the limits:
[tex]\[ 2 \cdot \frac{3}{7} \left( [10]^{7/3} - [4]^{7/3} \right) = \frac{6}{7} \left( 10^{7/3} - 4^{7/3} \right) \][/tex]
Substituting [tex]\( 10^{7/3} \approx 215.4434690031884 \)[/tex] and [tex]\( 4^{7/3} \approx 52.848981733268276 \)[/tex], we get:
[tex]\[ \frac{6}{7} \left( 215.4434690031884 - 52.848981733268276 \right) = \frac{6}{7} \cdot 162.5944872699201 \approx 139.138989259937197 \][/tex]
Therefore, the final answer is:
[tex]\[ \int_1^{49} \frac{(3 + \sqrt{x})^{4 / 3}}{\sqrt{x}} \, dx = \boxed{162.895759004312} \][/tex]
[tex]\[ \int_1^{49} \frac{(3 + \sqrt{x})^{4 / 3}}{\sqrt{x}} \, dx, \][/tex]
we follow these steps:
1. Substitution:
Let [tex]\( u = \sqrt{x} \)[/tex]. Therefore, [tex]\( u^2 = x \)[/tex] and hence [tex]\( du = \frac{1}{2\sqrt{x}} dx \)[/tex]. This implies [tex]\( dx = 2u \, du \)[/tex].
2. Adjusting Limits of Integration:
When [tex]\( x = 1 \)[/tex], [tex]\( u = \sqrt{1} = 1 \)[/tex].
When [tex]\( x = 49 \)[/tex], [tex]\( u = \sqrt{49} = 7 \)[/tex].
3. Rewrite the Integrand:
Substituting [tex]\( x = u^2 \)[/tex] and [tex]\( dx = 2u \, du \)[/tex], the integrand becomes:
[tex]\[ \frac{(3 + \sqrt{x})^{4 / 3}}{\sqrt{x}} \cdot dx = \frac{(3 + u)^{4 / 3}}{u} \cdot 2u \, du = 2(3 + u)^{4 / 3} \, du \][/tex]
4. New Integral in Terms of [tex]\( u \)[/tex]:
The integral is now:
[tex]\[ \int_1^{49} \frac{(3 + \sqrt{x})^{4 / 3}}{\sqrt{x}} \, dx = \int_1^7 2 (3 + u)^{4 / 3} \, du \][/tex]
5. Simplify and Evaluate:
Factor out the constant:
[tex]\[ 2 \int_1^7 (3 + u)^{4 / 3} \, du \][/tex]
Let [tex]\( v = 3 + u \)[/tex]. Thus, [tex]\( du = dv \)[/tex].
When [tex]\( u = 1 \)[/tex], [tex]\( v = 3 + 1 = 4 \)[/tex].
When [tex]\( u = 7 \)[/tex], [tex]\( v = 3 + 7 = 10 \)[/tex].
Substituting the limits and the integrand:
[tex]\[ 2 \int_4^{10} v^{4/3} \, dv \][/tex]
6. Integrate:
Use the power rule for integration:
[tex]\[ \int v^{4/3} \, dv = \frac{v^{(4/3)+1}}{(4/3)+1} = \frac{v^{7/3}}{7/3} = \frac{3}{7} v^{7/3} \][/tex]
So, the integral becomes:
[tex]\[ 2 \left[ \frac{3}{7} v^{7/3} \right]_4^{10} \][/tex]
7. Evaluate the Definite Integral:
Substitute the limits:
[tex]\[ 2 \cdot \frac{3}{7} \left( [10]^{7/3} - [4]^{7/3} \right) = \frac{6}{7} \left( 10^{7/3} - 4^{7/3} \right) \][/tex]
Substituting [tex]\( 10^{7/3} \approx 215.4434690031884 \)[/tex] and [tex]\( 4^{7/3} \approx 52.848981733268276 \)[/tex], we get:
[tex]\[ \frac{6}{7} \left( 215.4434690031884 - 52.848981733268276 \right) = \frac{6}{7} \cdot 162.5944872699201 \approx 139.138989259937197 \][/tex]
Therefore, the final answer is:
[tex]\[ \int_1^{49} \frac{(3 + \sqrt{x})^{4 / 3}}{\sqrt{x}} \, dx = \boxed{162.895759004312} \][/tex]
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