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Sagot :
To find the integral [tex]\(\int \frac{2x - 1}{4 \sqrt{x}} \, dx\)[/tex], we can approach the problem step-by-step.
1. Rewrite the integrand in a more manageable form:
The given integrand is [tex]\(\frac{2x - 1}{4 \sqrt{x}}\)[/tex]. We can split this into two separate fractions:
[tex]\[ \frac{2x - 1}{4 \sqrt{x}} = \frac{2x}{4 \sqrt{x}} - \frac{1}{4 \sqrt{x}} \][/tex]
Simplifying the fractions individually, we have:
[tex]\[ \frac{2x}{4 \sqrt{x}} = \frac{2}{4} \cdot \frac{x}{\sqrt{x}} = \frac{1}{2} \cdot \frac{x}{\sqrt{x}} = \frac{1}{2} \cdot \sqrt{x} = \frac{\sqrt{x}}{2} \][/tex]
And:
[tex]\[ \frac{1}{4 \sqrt{x}} = \frac{1}{4} \cdot x^{-\frac{1}{2}} = \frac{1}{4} x^{-\frac{1}{2}} \][/tex]
Therefore:
[tex]\[ \frac{2x - 1}{4 \sqrt{x}} = \frac{\sqrt{x}}{2} - \frac{1}{4} x^{-\frac{1}{2}} \][/tex]
2. Integrate each term separately:
We now have two simpler integrals to solve:
[tex]\[ \int \left( \frac{\sqrt{x}}{2} - \frac{1}{4} x^{-\frac{1}{2}} \right) dx \][/tex]
Start by integrating [tex]\(\frac{\sqrt{x}}{2}\)[/tex]:
[tex]\[ \frac{1}{2} \int \sqrt{x} \, dx = \frac{1}{2} \int x^{\frac{1}{2}} \, dx \][/tex]
Applying the power rule for integration [tex]\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)[/tex]:
[tex]\[ \frac{1}{2} \cdot \frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1} = \frac{1}{2} \cdot \frac{x^{\frac{3}{2}}}{\frac{3}{2}} = \frac{1}{2} \cdot \frac{2}{3} x^{\frac{3}{2}} = \frac{1}{3} x^{\frac{3}{2}} \][/tex]
Next, integrate [tex]\(\frac{1}{4} x^{-\frac{1}{2}}\)[/tex]:
[tex]\[ -\frac{1}{4} \int x^{-\frac{1}{2}} \, dx \][/tex]
Again, using the power rule:
[tex]\[ -\frac{1}{4} \cdot \frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1} = -\frac{1}{4} \cdot \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = -\frac{1}{4} \cdot 2 x^{\frac{1}{2}} = -\frac{1}{2} x^{\frac{1}{2}} \][/tex]
3. Combine the results:
Combining both integrals, we get:
[tex]\[ \frac{1}{3} x^{\frac{3}{2}} - \frac{1}{2} x^{\frac{1}{2}} \][/tex]
Simplifying and combining into a single expression:
[tex]\[ \frac{\sqrt{x} (2x - 3)}{6} \][/tex]
4. Include the constant of integration:
Finally, we add the constant of integration [tex]\(C\)[/tex] to the result:
[tex]\[ \int \frac{2x - 1}{4 \sqrt{x}} \, dx = \frac{\sqrt{x} (2x - 3)}{6} + C \][/tex]
So, the simplified form of the integral is:
[tex]\[ \boxed{\frac{\sqrt{x} (2x - 3)}{6} + C} \][/tex]
1. Rewrite the integrand in a more manageable form:
The given integrand is [tex]\(\frac{2x - 1}{4 \sqrt{x}}\)[/tex]. We can split this into two separate fractions:
[tex]\[ \frac{2x - 1}{4 \sqrt{x}} = \frac{2x}{4 \sqrt{x}} - \frac{1}{4 \sqrt{x}} \][/tex]
Simplifying the fractions individually, we have:
[tex]\[ \frac{2x}{4 \sqrt{x}} = \frac{2}{4} \cdot \frac{x}{\sqrt{x}} = \frac{1}{2} \cdot \frac{x}{\sqrt{x}} = \frac{1}{2} \cdot \sqrt{x} = \frac{\sqrt{x}}{2} \][/tex]
And:
[tex]\[ \frac{1}{4 \sqrt{x}} = \frac{1}{4} \cdot x^{-\frac{1}{2}} = \frac{1}{4} x^{-\frac{1}{2}} \][/tex]
Therefore:
[tex]\[ \frac{2x - 1}{4 \sqrt{x}} = \frac{\sqrt{x}}{2} - \frac{1}{4} x^{-\frac{1}{2}} \][/tex]
2. Integrate each term separately:
We now have two simpler integrals to solve:
[tex]\[ \int \left( \frac{\sqrt{x}}{2} - \frac{1}{4} x^{-\frac{1}{2}} \right) dx \][/tex]
Start by integrating [tex]\(\frac{\sqrt{x}}{2}\)[/tex]:
[tex]\[ \frac{1}{2} \int \sqrt{x} \, dx = \frac{1}{2} \int x^{\frac{1}{2}} \, dx \][/tex]
Applying the power rule for integration [tex]\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)[/tex]:
[tex]\[ \frac{1}{2} \cdot \frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1} = \frac{1}{2} \cdot \frac{x^{\frac{3}{2}}}{\frac{3}{2}} = \frac{1}{2} \cdot \frac{2}{3} x^{\frac{3}{2}} = \frac{1}{3} x^{\frac{3}{2}} \][/tex]
Next, integrate [tex]\(\frac{1}{4} x^{-\frac{1}{2}}\)[/tex]:
[tex]\[ -\frac{1}{4} \int x^{-\frac{1}{2}} \, dx \][/tex]
Again, using the power rule:
[tex]\[ -\frac{1}{4} \cdot \frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1} = -\frac{1}{4} \cdot \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = -\frac{1}{4} \cdot 2 x^{\frac{1}{2}} = -\frac{1}{2} x^{\frac{1}{2}} \][/tex]
3. Combine the results:
Combining both integrals, we get:
[tex]\[ \frac{1}{3} x^{\frac{3}{2}} - \frac{1}{2} x^{\frac{1}{2}} \][/tex]
Simplifying and combining into a single expression:
[tex]\[ \frac{\sqrt{x} (2x - 3)}{6} \][/tex]
4. Include the constant of integration:
Finally, we add the constant of integration [tex]\(C\)[/tex] to the result:
[tex]\[ \int \frac{2x - 1}{4 \sqrt{x}} \, dx = \frac{\sqrt{x} (2x - 3)}{6} + C \][/tex]
So, the simplified form of the integral is:
[tex]\[ \boxed{\frac{\sqrt{x} (2x - 3)}{6} + C} \][/tex]
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