How do you evaluate the definite integral #int (xsqrtx)dx# from [4,9]?

Question

Ezra Adams · Accepted Answer

The answer is #=422/5#

To evaluate this integral, we use

#intx^ndx=x^(n+1)/(n+1)+C#

#xsqrtx=x*x^(1/2)=x^(3/2)#

#int_4^9(xsqrtx)dx=int_4^9x^(3/2)dx= [x^(5/2)/(5/2) ]_4^9 #

#= [ 2/5x^(5/2) ]_4^9 #

#=2/5(3^5-2^5)#

#=(486-64)/5#

#=422/5#

Aurora Alvarado · Answer

#422/5# Express #xsqrtx=x xxx^(1/2)=x^(3/2)# #rArrint_4^9x^(3/2)dx# integrate using the #color(blue)"power rule for integration"# #color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(intax^ndx=a/(n+1)x^(n+1))color(white)(2/2)|)))# #rArrint_4^9x^(3/2)dx=[1/(5/2)x^(5/2)]_4^9=[2/5x^(5/2)]_4^9# #=2/5(9^(5/2)-4^(5/2))# #=2/5(243-32)=422/5#

Emilia Aguilar · Answer

undefined To evaluate the definite integral $ \int_{4}^{9} x \sqrt{x} \, dx $, we can use the integration by substitution method. Let $ u = \sqrt{x} $. Then, $ x = u^2 $ and $ dx = 2u \, du $. Substituting these into the integral, we get:

$$ \int_{4}^{9} x \sqrt{x} \, dx = \int_{2}^{3} u^2 \cdot u \cdot 2u \, du $$

$$ = 2 \int_{2}^{3} u^4 \, du $$

$$ = 2 \left[ \frac{u^5}{5} \right]_{2}^{3} $$

$$ = 2 \left( \frac{3^5}{5} - \frac{2^5}{5} \right) $$

$$ = 2 \left( \frac{243 - 32}{5} \right) $$

$$ = 2 \left( \frac{211}{5} \right) $$

$$ = \frac{422}{5} $$

So, the value of the definite integral $ \int_{4}^{9} x \sqrt{x} \, dx $ is $ \frac{422}{5} $.

Isaiah Allen · Answer

undefined To evaluate the definite integral $\int_{4}^{9} x \sqrt{x} \, dx$, you can use the power rule for integration and the properties of definite integrals. First, rewrite the integrand as $x^{\frac{3}{2}}$, then integrate using the power rule. Finally, evaluate the antiderivative at the upper and lower limits of integration and find the difference. Here's the calculation:

1. Integrate $x^{\frac{3}{2}}$ with respect to $x$:
$$ \int x^{\frac{3}{2}} \, dx = \frac{2}{5} x^{\frac{5}{2}} + C $$

2. Evaluate the antiderivative at the upper and lower limits:
$$ \left(\frac{2}{5} \cdot 9^{\frac{5}{2}}\right) - \left(\frac{2}{5} \cdot 4^{\frac{5}{2}}\right) $$

3. Simplify and compute the difference to find the value of the definite integral.