What is the antiderivative of #(t - 9t^2)/sqrt(t) dt#?

I got #(2/15)[t^(3/2)](5 - 27t) + C#. Is this correct?

Answer 1

# \ #

# \ 2/15 t^{3/2} ( 5 - 27 t ) + C. #

# \mbox{You Got It Exactly Right !! Bravo !!!} #

# \ #
# \mbox{This antiderivative allows a nice, quick rewrite of the integrand,} \ \ \mbox{which will then allow a thankfully smooth integration.} \ \ \mbox{Simplification later may be only slightly less smooth.} #
# \mbox{We have:} #
# \int \ \ { t - 9 t^2 } / \sqrt{t} dt \quad = \ \int \ \ ( t / \sqrt{t} - { 9 t ^2 } / \sqrt{t} ) dt #
# \qquad \qquad qquad \qquad qquad \qquad = \ \int \ \ ( t^{1/2} - 9 t^{3/2} ) dt #
# \qquad \qquad qquad \qquad qquad \qquad = \ 2/3 t^{3/2} - 9 ( 2/5 t^{5/2} ) + C. #
# \mbox{To simplify this now, factor out the numerical fractions using} \ \ \mbox{the LCM of their denominators and the GCD of their} \ \ \mbox{numerators, and factor out the common variables using their} \ \ \mbox{lowest common power: #
# \int \ \ { t - 9 t^2 } / \sqrt{t} dt \quad = 2/3 t^{3/2} - 9 ( 2/5 t^{5/2} ) + C #
# \qquad \qquad qquad \qquad qquad \qquad = \ 2/15 t^{3/2} (5) - \ 2/15 t^{3/2} ( 9 \cdot 3 t) + C. #
# \qquad \qquad qquad \qquad qquad \qquad = \ 2/15 t^{3/2} ( 5 - 27 t ) + C. #
# \ #
# \mbox{You Got It Exactly Right !! Bravo !!!} #
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Answer 2

#int (t-9t^2)/sqrtt dt=2/3t^(3/2)-18/5t^(5/2)+"c"#

#int (t-9t^2)/sqrtt dt = intt/sqrtt dt -9intt^2/sqrtt dt = intt^(1/2)dt -9int t^(3/2)dt#

Now, for integration, we apply the power rule:

#int x^n dx = 1/(n+1)x^(n+1)+"c"#
#therefore intt^(1/2)dt -9int t^(3/2)dt= 2/3t^(3/2)-18/5t^(5/2)+"c"#

To confirm, differentiating provides

#t^(1/2) -9t^(3/2) =(t-9t^2)/sqrtt#
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Answer 3

To find the antiderivative of ( \frac{t - 9t^2}{\sqrt{t}} , dt ), you can split the expression into two separate integrals:

[ \int \frac{t}{\sqrt{t}} , dt - \int \frac{9t^2}{\sqrt{t}} , dt ]

Then, you can use basic integration rules:

[ \int \frac{t}{\sqrt{t}} , dt = \int t^{\frac{1}{2}} , dt = \frac{2}{3}t^{\frac{3}{2}} + C ]

[ \int \frac{9t^2}{\sqrt{t}} , dt = 9 \int t^{\frac{5}{2}} , dt = \frac{18}{7}t^{\frac{7}{2}} + C ]

Where ( C ) is the constant of integration. So, the antiderivative of ( \frac{t - 9t^2}{\sqrt{t}} ) is:

[ \frac{2}{3}t^{\frac{3}{2}} - \frac{18}{7}t^{\frac{7}{2}} + C ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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