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A integral definida de uma função f (x), num intervalo [a, b] é igual à área entre a curva de f(x) e o eixo dos x, que é o mesmo resultado obtido por integração. Calcule as seguintes integrais definidas:


To solve the given definite integral 

\[
\int_{1}^{4} \left( \frac{3}{\sqrt{t}} - 5\sqrt{t} - \frac{1}{\sqrt{t^3}} \right) dt,
\]

we will integrate each term separately.

First, rewrite the integral in a simpler form by expressing each term with exponents:

\[
\int_{1}^{4} \left( 3t^{-\frac{1}{2}} - 5t^{\frac{1}{2}} - t^{-\frac{3}{2}} \right) dt.
\]

Now integrate each term individually:

1. \(\int 3t^{-\frac{1}{2}} \, dt\):
   \[
   \int 3t^{-\frac{1}{2}} \, dt = 3 \int t^{-\frac{1}{2}} \, dt = 3 \cdot 2t^{\frac{1}{2

Learn how to calculate definite integrals using step-by-step solutions. This problem involves computing the definite integral of a given function over a specified interval. Understand the concept of the area under the curve and apply the Fundamental Theorem of Calculus. Suitable for undergraduate students and anyone interested in integral calculus.

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