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

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

I got

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Now, for integration, we apply the power rule:

To confirm, differentiating provides

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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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