IDNLearn.com offers a unique blend of expert answers and community-driven insights. Get thorough and trustworthy answers to your queries from our extensive network of knowledgeable professionals.
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]
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com has the answers you need. Thank you for visiting, and we look forward to helping you again soon.
