Answer :
Certainly! Let's work through the problem step-by-step to evaluate the integral [tex]\(\int \frac{3}{4} x^{\frac{1}{2}} \, dx\)[/tex].
1. Identify the integral to solve:
[tex]\[ \int \frac{3}{4} x^{\frac{1}{2}} \, dx \][/tex]
2. Rewrite the integrand for easier manipulation:
The integrand can be rewritten in a more convenient form. Notice that [tex]\( x^{\frac{1}{2}} \)[/tex] is the same as [tex]\( \sqrt{x} \)[/tex]. We can express the integral as:
[tex]\[ \int \frac{3}{4} x^{\frac{1}{2}} \, dx \][/tex]
3. Factor out the constant:
The constant [tex]\(\frac{3}{4}\)[/tex] can be factored out of the integral:
[tex]\[ \frac{3}{4} \int x^{\frac{1}{2}} \, dx \][/tex]
4. Use the power rule for integration:
Recall the power rule for integration, which states that [tex]\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)[/tex], where [tex]\(n \neq -1\)[/tex]. Here, our exponent [tex]\( n \)[/tex] is [tex]\(\frac{1}{2}\)[/tex]. Applying the power rule:
[tex]\[ \int x^{\frac{1}{2}} \, dx = \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} + C = \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + C \][/tex]
5. Simplify the result:
Simplify [tex]\(\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\)[/tex] by multiplying the numerator by the reciprocal of the denominator:
[tex]\[ \frac{x^{\frac{3}{2}}}{\frac{3}{2}} = x^{\frac{3}{2}} \times \frac{2}{3} = \frac{2}{3} x^{\frac{3}{2}} \][/tex]
6. Multiply back the constant factor:
Remember that we factored out the [tex]\(\frac{3}{4}\)[/tex] earlier, so we need to multiply it back:
[tex]\[ \frac{3}{4} \times \left( \frac{2}{3} x^{\frac{3}{2}} \right) \][/tex]
7. Simplify the expression:
Simplify the constants:
[tex]\[ \frac{3}{4} \times \frac{2}{3} x^{\frac{3}{2}} = \frac{3 \times 2}{4 \times 3} x^{\frac{3}{2}} = \frac{1}{2} x^{\frac{3}{2}} \][/tex]
8. Include the constant of integration:
Don't forget to add the constant of integration [tex]\(C\)[/tex]:
[tex]\[ \frac{1}{2} x^{\frac{3}{2}} + C \][/tex]
So, the evaluated integral is:
[tex]\[ \int \frac{3}{4} x^{\frac{1}{2}} \, dx = \frac{1}{2} x^{\frac{3}{2}} + C \][/tex]
This is the detailed, step-by-step solution to the integral [tex]\(\int \frac{3}{4} x^{\frac{1}{2}} \, dx\)[/tex].
1. Identify the integral to solve:
[tex]\[ \int \frac{3}{4} x^{\frac{1}{2}} \, dx \][/tex]
2. Rewrite the integrand for easier manipulation:
The integrand can be rewritten in a more convenient form. Notice that [tex]\( x^{\frac{1}{2}} \)[/tex] is the same as [tex]\( \sqrt{x} \)[/tex]. We can express the integral as:
[tex]\[ \int \frac{3}{4} x^{\frac{1}{2}} \, dx \][/tex]
3. Factor out the constant:
The constant [tex]\(\frac{3}{4}\)[/tex] can be factored out of the integral:
[tex]\[ \frac{3}{4} \int x^{\frac{1}{2}} \, dx \][/tex]
4. Use the power rule for integration:
Recall the power rule for integration, which states that [tex]\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)[/tex], where [tex]\(n \neq -1\)[/tex]. Here, our exponent [tex]\( n \)[/tex] is [tex]\(\frac{1}{2}\)[/tex]. Applying the power rule:
[tex]\[ \int x^{\frac{1}{2}} \, dx = \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} + C = \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + C \][/tex]
5. Simplify the result:
Simplify [tex]\(\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\)[/tex] by multiplying the numerator by the reciprocal of the denominator:
[tex]\[ \frac{x^{\frac{3}{2}}}{\frac{3}{2}} = x^{\frac{3}{2}} \times \frac{2}{3} = \frac{2}{3} x^{\frac{3}{2}} \][/tex]
6. Multiply back the constant factor:
Remember that we factored out the [tex]\(\frac{3}{4}\)[/tex] earlier, so we need to multiply it back:
[tex]\[ \frac{3}{4} \times \left( \frac{2}{3} x^{\frac{3}{2}} \right) \][/tex]
7. Simplify the expression:
Simplify the constants:
[tex]\[ \frac{3}{4} \times \frac{2}{3} x^{\frac{3}{2}} = \frac{3 \times 2}{4 \times 3} x^{\frac{3}{2}} = \frac{1}{2} x^{\frac{3}{2}} \][/tex]
8. Include the constant of integration:
Don't forget to add the constant of integration [tex]\(C\)[/tex]:
[tex]\[ \frac{1}{2} x^{\frac{3}{2}} + C \][/tex]
So, the evaluated integral is:
[tex]\[ \int \frac{3}{4} x^{\frac{1}{2}} \, dx = \frac{1}{2} x^{\frac{3}{2}} + C \][/tex]
This is the detailed, step-by-step solution to the integral [tex]\(\int \frac{3}{4} x^{\frac{1}{2}} \, dx\)[/tex].
