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Sagot :
Sure! Let's break down the problem step by step to find the expression for [tex]\( f(x) \)[/tex].
1. Integrate [tex]\( f'(x) \)[/tex] to find [tex]\( f(x) \)[/tex]:
We're given that [tex]\( f'(x) = (5x + 2)^{-\frac{2}{3}} \)[/tex]. To find [tex]\( f(x) \)[/tex], we need to integrate [tex]\( f'(x) \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ f(x) = \int (5x + 2)^{-\frac{2}{3}} \, dx \][/tex]
2. Use substitution to simplify the integral:
Let's perform a substitution to simplify the integration. Let [tex]\( u = 5x + 2 \)[/tex]. Then, [tex]\( du = 5dx \)[/tex] or [tex]\( dx = \frac{1}{5}du \)[/tex].
Substitute back into the integral:
[tex]\[ f(x) = \int (u)^{-\frac{2}{3}} \cdot \frac{1}{5} \, du \][/tex]
[tex]\[ f(x) = \frac{1}{5} \int u^{-\frac{2}{3}} \, du \][/tex]
3. Integrate with respect to [tex]\( u \)[/tex]:
The integral of [tex]\( u^{-\frac{2}{3}} \)[/tex] is found using the power rule for integration. Recall that [tex]\( \int u^n \, du = \frac{u^{n+1}}{n+1} \)[/tex], where [tex]\( n \neq -1 \)[/tex].
In this case, [tex]\( n = -\frac{2}{3} \)[/tex]:
[tex]\[ \frac{1}{5} \int u^{-\frac{2}{3}} \, du = \frac{1}{5} \cdot \frac{u^{-\frac{2}{3} + 1}}{-\frac{2}{3} + 1} = \frac{1}{5} \cdot \frac{u^{\frac{1}{3}}}{\frac{1}{3}} \][/tex]
Simplify the expression:
[tex]\[ \frac{1}{5} \cdot 3u^{\frac{1}{3}} = \frac{3}{5} u^{\frac{1}{3}} \][/tex]
4. Substitute back for [tex]\( u \)[/tex]:
Recall that [tex]\( u = 5x + 2 \)[/tex]:
[tex]\[ f(x) = \frac{3}{5} (5x + 2)^{\frac{1}{3}} + C \][/tex]
5. Use the initial condition to solve for [tex]\( C \)[/tex]:
We are given that [tex]\( f(6) = \frac{26}{3} \)[/tex]. Substitute [tex]\( x = 6 \)[/tex] and [tex]\( f(x) = \frac{26}{3} \)[/tex] into the equation:
[tex]\[ \frac{26}{3} = \frac{3}{5} (5 \cdot 6 + 2)^{\frac{1}{3}} + C \][/tex]
Simplify inside the parentheses:
[tex]\[ \frac{26}{3} = \frac{3}{5} (30 + 2)^{\frac{1}{3}} + C \][/tex]
[tex]\[ \frac{26}{3} = \frac{3}{5} \cdot 32^{\frac{1}{3}} + C \][/tex]
Since [tex]\( 32^{\frac{1}{3}} = 2 \)[/tex]:
[tex]\[ \frac{26}{3} = \frac{3}{5} \cdot 2 + C \][/tex]
Simplify further:
[tex]\[ \frac{26}{3} = \frac{6}{5} + C \][/tex]
Solve for [tex]\( C \)[/tex]:
[tex]\[ \frac{26}{3} - \frac{6}{5} = C \][/tex]
Find a common denominator:
[tex]\[ \frac{26 \cdot 5}{3 \cdot 5} - \frac{6 \cdot 3}{5 \cdot 3} = C \][/tex]
[tex]\[ \frac{130}{15} - \frac{18}{15} = C \][/tex]
[tex]\[ \frac{112}{15} = C \][/tex]
6. Write the final expression for [tex]\( f(x) \)[/tex]:
The expression for [tex]\( f(x) \)[/tex] is:
[tex]\[ f(x) = \frac{3}{5} (5x + 2)^{\frac{1}{3}} + \frac{112}{15} \][/tex]
Thus, the final expression for [tex]\( f(x) \)[/tex] is:
[tex]\[ f(x) = \frac{3}{5} (5x + 2)^{\frac{1}{3}} + \frac{112}{15} \][/tex]
1. Integrate [tex]\( f'(x) \)[/tex] to find [tex]\( f(x) \)[/tex]:
We're given that [tex]\( f'(x) = (5x + 2)^{-\frac{2}{3}} \)[/tex]. To find [tex]\( f(x) \)[/tex], we need to integrate [tex]\( f'(x) \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ f(x) = \int (5x + 2)^{-\frac{2}{3}} \, dx \][/tex]
2. Use substitution to simplify the integral:
Let's perform a substitution to simplify the integration. Let [tex]\( u = 5x + 2 \)[/tex]. Then, [tex]\( du = 5dx \)[/tex] or [tex]\( dx = \frac{1}{5}du \)[/tex].
Substitute back into the integral:
[tex]\[ f(x) = \int (u)^{-\frac{2}{3}} \cdot \frac{1}{5} \, du \][/tex]
[tex]\[ f(x) = \frac{1}{5} \int u^{-\frac{2}{3}} \, du \][/tex]
3. Integrate with respect to [tex]\( u \)[/tex]:
The integral of [tex]\( u^{-\frac{2}{3}} \)[/tex] is found using the power rule for integration. Recall that [tex]\( \int u^n \, du = \frac{u^{n+1}}{n+1} \)[/tex], where [tex]\( n \neq -1 \)[/tex].
In this case, [tex]\( n = -\frac{2}{3} \)[/tex]:
[tex]\[ \frac{1}{5} \int u^{-\frac{2}{3}} \, du = \frac{1}{5} \cdot \frac{u^{-\frac{2}{3} + 1}}{-\frac{2}{3} + 1} = \frac{1}{5} \cdot \frac{u^{\frac{1}{3}}}{\frac{1}{3}} \][/tex]
Simplify the expression:
[tex]\[ \frac{1}{5} \cdot 3u^{\frac{1}{3}} = \frac{3}{5} u^{\frac{1}{3}} \][/tex]
4. Substitute back for [tex]\( u \)[/tex]:
Recall that [tex]\( u = 5x + 2 \)[/tex]:
[tex]\[ f(x) = \frac{3}{5} (5x + 2)^{\frac{1}{3}} + C \][/tex]
5. Use the initial condition to solve for [tex]\( C \)[/tex]:
We are given that [tex]\( f(6) = \frac{26}{3} \)[/tex]. Substitute [tex]\( x = 6 \)[/tex] and [tex]\( f(x) = \frac{26}{3} \)[/tex] into the equation:
[tex]\[ \frac{26}{3} = \frac{3}{5} (5 \cdot 6 + 2)^{\frac{1}{3}} + C \][/tex]
Simplify inside the parentheses:
[tex]\[ \frac{26}{3} = \frac{3}{5} (30 + 2)^{\frac{1}{3}} + C \][/tex]
[tex]\[ \frac{26}{3} = \frac{3}{5} \cdot 32^{\frac{1}{3}} + C \][/tex]
Since [tex]\( 32^{\frac{1}{3}} = 2 \)[/tex]:
[tex]\[ \frac{26}{3} = \frac{3}{5} \cdot 2 + C \][/tex]
Simplify further:
[tex]\[ \frac{26}{3} = \frac{6}{5} + C \][/tex]
Solve for [tex]\( C \)[/tex]:
[tex]\[ \frac{26}{3} - \frac{6}{5} = C \][/tex]
Find a common denominator:
[tex]\[ \frac{26 \cdot 5}{3 \cdot 5} - \frac{6 \cdot 3}{5 \cdot 3} = C \][/tex]
[tex]\[ \frac{130}{15} - \frac{18}{15} = C \][/tex]
[tex]\[ \frac{112}{15} = C \][/tex]
6. Write the final expression for [tex]\( f(x) \)[/tex]:
The expression for [tex]\( f(x) \)[/tex] is:
[tex]\[ f(x) = \frac{3}{5} (5x + 2)^{\frac{1}{3}} + \frac{112}{15} \][/tex]
Thus, the final expression for [tex]\( f(x) \)[/tex] is:
[tex]\[ f(x) = \frac{3}{5} (5x + 2)^{\frac{1}{3}} + \frac{112}{15} \][/tex]
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