To solve the integral [tex]\(\int x^{\frac{3}{4}} \, dx\)[/tex], we will follow the steps of indefinite integration using the power rule.
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Step 1: Identify the function to integrate
The integrand is [tex]\(x^{\frac{3}{4}}\)[/tex].
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Step 2: Apply the power rule for integration
The power rule for integration states that for any function of the form [tex]\(x^n\)[/tex] where [tex]\(n \neq -1\)[/tex]:
[tex]\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
In this problem, our exponent [tex]\(n\)[/tex] is [tex]\(\frac{3}{4}\)[/tex].
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Step 3: Compute [tex]\(n + 1\)[/tex]
First, we add 1 to the exponent [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[
n + 1 = \frac{3}{4} + 1 = \frac{3}{4} + \frac{4}{4} = \frac{7}{4}
\][/tex]
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Step 4: Apply the power rule
Now substitute [tex]\(n = \frac{3}{4}\)[/tex] and [tex]\(n + 1 = \frac{7}{4}\)[/tex] into the power rule formula:
[tex]\[
\int x^{\frac{3}{4}} \, dx = \frac{x^{\frac{7}{4}}}{\frac{7}{4}} + C
\][/tex]
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Step 5: Simplify the result
To simplify the fraction [tex]\(\frac{x^{\frac{7}{4}}}{\frac{7}{4}}\)[/tex], we multiply by the reciprocal of [tex]\(\frac{7}{4}\)[/tex]:
[tex]\[
\frac{x^{\frac{7}{4}}}{\frac{7}{4}} = x^{\frac{7}{4}} \times \frac{4}{7} = \frac{4}{7} x^{\frac{7}{4}}
\][/tex]
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Step 6: Express the answer
Finally, we express the integral with the simplified coefficient and add the constant of integration [tex]\(C\)[/tex]:
[tex]\[
\int x^{\frac{3}{4}} \, dx = \frac{4}{7} x^{\frac{7}{4}} + C
\][/tex]
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Therefore, our solution to the integral [tex]\(\int x^{\frac{3}{4}} \, dx\)[/tex] is:
[tex]\[
\boxed{0.571428571428571 \, x^{1.75} + C}
\][/tex]
