To solve the integral \(\int\left(2 \sqrt{t}-t-\frac{9}{t^{2}}\right) d t\), we can break it down into three separate integrals: \(\int 2 \sqrt{t} \, dt\), \(\int -t \, dt\), and \(\int -\frac{9}{t^2} \, dt\). Each of these can be integrated using basic integration rules. The first term involves a power of \(t\), the second is a simple polynomial, and the third is a power of \(t\) with a negative exponent.
Consideramos la integral \(\int\left(2 \sqrt{t}-t-\frac{9}{t^{2}}\right) d t\). Esta integral se puede descomponer en tres partes:
\[
\int 2 \sqrt{t} \, dt, \quad \int -t \, dt, \quad \int -\frac{9}{t^2} \, dt
\]
Calculamos cada una de las integrales por separado:
Para \(\int 2 \sqrt{t} \, dt\):
\[
\int 2 \sqrt{t} \, dt = \frac{4}{3} t^{3/2}
\]
Para \(\int -t \, dt\):
\[
\int -t \, dt = -\frac{t^2}{2}
\]
Para \(\int -\frac{9}{t^2} \, dt\):
\[
\int -\frac{9}{t^2} \, dt = 9 \cdot \frac{1}{t} = \frac{9}{t}
\]
Sumamos los resultados de las integrales:
\[
\int\left(2 \sqrt{t}-t-\frac{9}{t^{2}}\right) d t = \frac{4}{3} t^{3/2} - \frac{t^2}{2} + \frac{9}{t} + C
\]
donde \(C\) es la constante de integración.
\[
\boxed{\int\left(2 \sqrt{t}-t-\frac{9}{t^{2}}\right) d t = \frac{4}{3} t^{3/2} - \frac{t^2}{2} + \frac{9}{t} + C}
\]
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