Understanding the Integral of ( x^{3/4} )
Calculating the integral of a function is a fundamental concept in calculus which helps us find the area under a curve, among other applications. In this article, we will explore the integration of the function ( f(x) = x^{3/4} ).
Integral Definition
The integral of a function ( f(x) ) is denoted as:
$\int f(x) , dx$
For our specific case, we want to compute:
$\int x^{3/4} , dx$
Power Rule of Integration
To solve this integral, we apply the power rule of integration. The power rule states that for any real number ( n \neq -1 ):
$\int x^n , dx = \frac{x^{n+1}}{n+1} + C$
where ( C ) is the constant of integration.
Applying the Power Rule
In our case, ( n = \frac{3}{4} ). We first calculate ( n + 1 ):
$n + 1 = \frac{3}{4} + 1 = \frac{3}{4} + \frac{4}{4} = \frac{7}{4}$
Now we can substitute ( n ) into the integral formula:
$\int x^{3/4} , dx = \frac{x^{7/4}}{7/4} + C$
To simplify this expression, we multiply by the reciprocal of ( \frac{7}{4} ):
$\int x^{3/4} , dx = \frac{4}{7} x^{7/4} + C$
Final Result
Thus, the integral of ( x^{3/4} ) is:
$\int x^{3/4} , dx = \frac{4}{7} x^{7/4} + C$
where ( C ) represents the arbitrary constant of integration, which reflects that there are infinitely many antiderivatives for every function.
Applications
The integral we derived can be applied in various fields such as physics, engineering, and economics, particularly when dealing with problems related to area calculation, volume of solids, and rate of change over time.
Conclusion
Understanding the integral of basic power functions like ( x^{3/4} ) enhances our ability to tackle more complex problems in calculus. By practicing the power rule and familiarizing ourselves with integration techniques, we become more proficient in mathematical analysis and application.
In summary, the integral of ( x^{3/4} ) is:
$\int x^{3/4} , dx = \frac{4}{7} x^{7/4} + C$
This provides a solid basis for further exploration into integration and its various applications.
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