Math Problem Statement

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:

Solution

To solve the given definite integral

14(3t5t1t3)dt,\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:

14(3t125t12t32)dt.\int_{1}^{4} \left( 3t^{-\frac{1}{2}} - 5t^{\frac{1}{2}} - t^{-\frac{3}{2}} \right) dt.

Now integrate each term individually:

  1. 3t12dt\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

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Math Problem Analysis

Mathematical Concepts

Definite Integral
Area under Curve
Integration

Formulas

Fundamental Theorem of Calculus

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Undergraduate