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Integrate the function $\frac{\left(5x-3\right)^3}{x^3}$ from $1$ to $\infty $

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Solution

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Final answer to the problem

The integral diverges.

Step-by-step Solution

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Expand the fraction $\frac{\left(5x\right)^3-9\left(5x\right)^2+135x-27}{x^3}$ into $4$ simpler fractions with common denominator $x^3$

$\int\left(\frac{\left(5x\right)^3}{x^3}+\frac{-9\left(5x\right)^2}{x^3}+\frac{135x}{x^3}+\frac{-27}{x^3}\right)dx$

Learn how to solve sum rule of differentiation problems step by step online.

$\int\left(\frac{\left(5x\right)^3}{x^3}+\frac{-9\left(5x\right)^2}{x^3}+\frac{135x}{x^3}+\frac{-27}{x^3}\right)dx$

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Learn how to solve sum rule of differentiation problems step by step online. Integrate the function ((5x-3)^3)/(x^3) from 1 to infinity. Expand the fraction \frac{\left(5x\right)^3-9\left(5x\right)^2+135x-27}{x^3} into 4 simpler fractions with common denominator x^3. Simplify the resulting fractions. Simplify the expression inside the integral. The integral \int\frac{\left(5x\right)^3}{x^3}dx results in: 125x.

Final answer to the problem

The integral diverges.

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Solve integral of (((5x-3)^3)/(x^3))dx from 1 to infinity using basic integralsSolve integral of (((5x-3)^3)/(x^3))dx from 1 to infinity using u-substitutionSolve integral of (((5x-3)^3)/(x^3))dx from 1 to infinity using integration by partsSolve integral of (((5x-3)^3)/(x^3))dx from 1 to infinity using trigonometric substitution

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Function Plot

Plotting: $\frac{\left(5x-3\right)^3}{x^3}$

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Main Topic: Sum Rule of Differentiation

The sum rule is a method to find the derivative of a function that is the sum of two or more functions.

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