Integration of Complex Functions: Detailed Steps for ∫(3x^2 - x1)/(x-1) dx

When dealing with integration in calculus, encountering complex functions like int;(3x^2 - x1)/(x-1) dx can be challenging. In this comprehensive guide, we will break down the steps to solve this integral, ensuring a clear understanding and adherence to best practices for search engines like Google.

Understanding the Integral

The given integral is:

I  ∫(3x^2 - x1)/(x-1) dx

By observing the function, we can split it into simpler components:

I  ∫(3x^2 - 3x   3) / (x-1) dx

Let's break this down further:

I  ∫(3x^2 / (x-1)) dx - ∫(3x / (x-1)) dx   ∫(3 / (x-1)) dx

Breaking Down the Integral

Now, let's integrate each term separately:

1. Integrating 3x^2 / (x-1)

∫(3x^2 / (x-1)) dx  3 ∫(x^2 / (x-1)) dx

Since x^2 / (x-1) can be decomposed using polynomial long division or partial fractions, we get:

x^2 / (x-1)  (x   1)   1/(x-1)

Thus:

∫(3x^2 / (x-1)) dx  3 ∫(x   1) dx   3 ∫(1 / (x-1)) dx

Integrating each term:

 3∫xdx   3∫1dx   3∫(1 / (x-1)) dx

Which results in:

 3(x^2/2)   3x   3 ln|x-1|   C1

2. Integrating 3x / (x-1)

∫(3x / (x-1)) dx  3 ∫(x / (x-1)) dx

Using polynomial long division, we get:

x / (x-1)  1   1/(x-1)

Thus:

∫(3x / (x-1)) dx  3 ∫(1   1 / (x-1)) dx

Integrating each term:

 3 ∫1dx   3∫(1 / (x-1)) dx

Which results in:

 3x   3 ln|x-1|   C2

3. Integrating 3 / (x-1)

∫(3 / (x-1)) dx  3∫(1 / (x-1)) dx

Integrating the term:

 3 ln|x-1|   C3

Combining the Results

Now, combining all the results:

I  (3(x^2/2)   3x   3 ln|x-1|) - (3x   3 ln|x-1|)   3 ln|x-1|   C

Simplifying:

I  3(x^2/2)   3 ln|x-1|   C

Where C is the constant of integration.

This final result is:

I  3(x^2/2)   3 ln|x-1|   C

Conclusion

This step-by-step approach not only simplifies the integration process but also ensures clarity and accuracy. By breaking down complex functions and integrating each component separately, we can solve even the most challenging integrals. This method is particularly useful for students and professionals in calculus and related fields.

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