Certainly! Let's go through the steps to evaluate the integral [tex]\( \int_0^1 x^{9/10} \, dx \)[/tex].
### Step 1: Determine the antiderivative
An antiderivative of [tex]\( x^n \)[/tex], as long as [tex]\( n \neq -1 \)[/tex], is given by:
[tex]\[ \frac{x^{n+1}}{n+1} \][/tex]
### Step 2: Apply the formula to find the antiderivative
Here, the exponent [tex]\( n = \frac{9}{10} \)[/tex]. Therefore, the antiderivative of [tex]\( x^{9/10} \)[/tex] is:
[tex]\[ \frac{x^{(9/10) + 1}}{(9/10) + 1} \][/tex]
Simplify [tex]\( (9/10) + 1 \)[/tex]:
[tex]\[ (9/10) + 1 = \frac{9}{10} + \frac{10}{10} = \frac{19}{10} \][/tex]
Thus, the antiderivative becomes:
[tex]\[ \frac{x^{19/10}}{19/10} \][/tex]
### Step 3: Simplify the antiderivative
To simplify further, we can rewrite [tex]\( \frac{1}{19/10} \)[/tex] as [tex]\( \frac{10}{19} \)[/tex]:
[tex]\[ \frac{x^{19/10}}{19/10} = \frac{10}{19} x^{19/10} \][/tex]
### Step 4: Evaluate the definite integral
We now need to evaluate the definite integral from 0 to 1:
[tex]\[ \left. \frac{10}{19} x^{19/10} \right|_0^1 \][/tex]
This means we must compute the value of the antiderivative at the upper and lower bounds and subtract:
[tex]\[ \left( \frac{10}{19} \cdot 1^{19/10} \right) - \left( \frac{10}{19} \cdot 0^{19/10} \right) \][/tex]
Substitute the bounds:
[tex]\[ \left( \frac{10}{19} \cdot 1 \right) - \left( \frac{10}{19} \cdot 0 \right) \][/tex]
### Step 5: Perform the final calculations
[tex]\[ \frac{10}{19} \cdot 1 = \frac{10}{19} \][/tex]
[tex]\[ \frac{10}{19} \cdot 0 = 0 \][/tex]
So, the definite integral evaluates to:
[tex]\[ \frac{10}{19} - 0 = \frac{10}{19} \][/tex]
### Final Answer
Converting [tex]\(\frac{10}{19}\)[/tex] to its decimal form:
[tex]\[ \frac{10}{19} \approx 0.5263157894736842 \][/tex]
Therefore, the value of the definite integral [tex]\(\int_0^1 x^{9/10} \, dx\)[/tex] is approximately [tex]\( 0.5263157894736842 \)[/tex].
