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Sagot :
To find the exact value of the integral [tex]\(\int_{-2}^6 \frac{4}{\sqrt[3]{x^2}} \, dx\)[/tex], let's follow these steps:
1. Rewrite the integrand:
The given integral is:
[tex]\[ \int_{-2}^6 \frac{4}{\sqrt[3]{x^2}} \, dx \][/tex]
Notice that [tex]\(\sqrt[3]{x^2}\)[/tex] can be rewritten using exponent notation:
[tex]\[ \sqrt[3]{x^2} = (x^2)^{1/3} = x^{2/3} \][/tex]
Therefore, the integrand can be rewritten as:
[tex]\[ \frac{4}{x^{2/3}} \][/tex]
This can be further simplified to:
[tex]\[ 4 x^{-2/3} \][/tex]
2. Set up the integral:
The integral now becomes:
[tex]\[ \int_{-2}^6 4 x^{-2/3} \, dx \][/tex]
3. Apply the power rule for integration:
To integrate [tex]\(x^n\)[/tex], where [tex]\(n \neq -1\)[/tex], we use the formula:
[tex]\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \][/tex]
In our case, [tex]\(n = -\frac{2}{3}\)[/tex]. Applying the rule, we get:
[tex]\[ \int x^{-\frac{2}{3}} \, dx = \frac{x^{1 - \frac{2}{3}}}{1 - \frac{2}{3}} + C = \frac{x^{\frac{1}{3}}}{\frac{1}{3}} + C = 3 x^{\frac{1}{3}} + C \][/tex]
Since our integrand is [tex]\(4 x^{-\frac{2}{3}}\)[/tex], we multiply the above result by 4:
[tex]\[ \int 4 x^{-\frac{2}{3}} \, dx = 4 \left( 3 x^{\frac{1}{3}} \right) + C = 12 x^{\frac{1}{3}} + C \][/tex]
4. Evaluate the definite integral:
We now need to evaluate this antiderivative over the interval [tex]\([-2, 6]\)[/tex]:
[tex]\[ \left[ 12 x^{\frac{1}{3}} \right]_{-2}^{6} \][/tex]
Evaluate at the upper limit [tex]\(x = 6\)[/tex]:
[tex]\[ 12 (6^{\frac{1}{3}}) \][/tex]
Evaluate at the lower limit [tex]\(x = -2\)[/tex]:
[tex]\[ 12 ((-2)^{\frac{1}{3}}) \][/tex]
Combining these, we get:
[tex]\[ 12 (6^{\frac{1}{3}}) - 12 ((-2)^{\frac{1}{3}}) \][/tex]
5. Simplify the expressions:
Calculate [tex]\(6^{\frac{1}{3}}\)[/tex] and [tex]\((-2)^{\frac{1}{3}}\)[/tex]:
[tex]\[ 12 (6^{\frac{1}{3}}) - 12 ((-2)^{\frac{1}{3}}) \][/tex]
These calculations result in:
[tex]\[ 12 \cdot 1.81712 - 12 \cdot (-1.25992) \][/tex]
[tex]\[ 12 \cdot 1.81712 + 12 \cdot 1.25992 = 21.8054 + 15.119 = 36.9244 \][/tex]
Hence, the exact value of the integral [tex]\(\int_{-2}^6 \frac{4}{\sqrt[3]{x^2}} \, dx\)[/tex] is:
[tex]\[ \boxed{36.9244997127241} \][/tex]
1. Rewrite the integrand:
The given integral is:
[tex]\[ \int_{-2}^6 \frac{4}{\sqrt[3]{x^2}} \, dx \][/tex]
Notice that [tex]\(\sqrt[3]{x^2}\)[/tex] can be rewritten using exponent notation:
[tex]\[ \sqrt[3]{x^2} = (x^2)^{1/3} = x^{2/3} \][/tex]
Therefore, the integrand can be rewritten as:
[tex]\[ \frac{4}{x^{2/3}} \][/tex]
This can be further simplified to:
[tex]\[ 4 x^{-2/3} \][/tex]
2. Set up the integral:
The integral now becomes:
[tex]\[ \int_{-2}^6 4 x^{-2/3} \, dx \][/tex]
3. Apply the power rule for integration:
To integrate [tex]\(x^n\)[/tex], where [tex]\(n \neq -1\)[/tex], we use the formula:
[tex]\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \][/tex]
In our case, [tex]\(n = -\frac{2}{3}\)[/tex]. Applying the rule, we get:
[tex]\[ \int x^{-\frac{2}{3}} \, dx = \frac{x^{1 - \frac{2}{3}}}{1 - \frac{2}{3}} + C = \frac{x^{\frac{1}{3}}}{\frac{1}{3}} + C = 3 x^{\frac{1}{3}} + C \][/tex]
Since our integrand is [tex]\(4 x^{-\frac{2}{3}}\)[/tex], we multiply the above result by 4:
[tex]\[ \int 4 x^{-\frac{2}{3}} \, dx = 4 \left( 3 x^{\frac{1}{3}} \right) + C = 12 x^{\frac{1}{3}} + C \][/tex]
4. Evaluate the definite integral:
We now need to evaluate this antiderivative over the interval [tex]\([-2, 6]\)[/tex]:
[tex]\[ \left[ 12 x^{\frac{1}{3}} \right]_{-2}^{6} \][/tex]
Evaluate at the upper limit [tex]\(x = 6\)[/tex]:
[tex]\[ 12 (6^{\frac{1}{3}}) \][/tex]
Evaluate at the lower limit [tex]\(x = -2\)[/tex]:
[tex]\[ 12 ((-2)^{\frac{1}{3}}) \][/tex]
Combining these, we get:
[tex]\[ 12 (6^{\frac{1}{3}}) - 12 ((-2)^{\frac{1}{3}}) \][/tex]
5. Simplify the expressions:
Calculate [tex]\(6^{\frac{1}{3}}\)[/tex] and [tex]\((-2)^{\frac{1}{3}}\)[/tex]:
[tex]\[ 12 (6^{\frac{1}{3}}) - 12 ((-2)^{\frac{1}{3}}) \][/tex]
These calculations result in:
[tex]\[ 12 \cdot 1.81712 - 12 \cdot (-1.25992) \][/tex]
[tex]\[ 12 \cdot 1.81712 + 12 \cdot 1.25992 = 21.8054 + 15.119 = 36.9244 \][/tex]
Hence, the exact value of the integral [tex]\(\int_{-2}^6 \frac{4}{\sqrt[3]{x^2}} \, dx\)[/tex] is:
[tex]\[ \boxed{36.9244997127241} \][/tex]
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