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Chapter 5: Problem 61

Using Technology to Find an Integral In Exercises \(57-62\) use a computer algebra system to find or evaluate the integral. $$ \int \frac{x^{2}}{x-1} d x $$

Short Answer

Expert verified
The result of the integral \( \int \frac{x^{2}}{x-1} \, dx \) is \( \frac{1}{2}x^2 + x + C \).

Step by step solution

01

Divide the Integral Using Polynomial Long Division

Begin the task by dividing \(x^2\) by \(x - 1\). This can be done manually or using a software tool. The result of this operation is \(x + 1\).
02

Reformulate the Integral

Now that the fraction has been divided, reformulate the integral as follows: \( \int (x + 1) \, dx \).
03

Integrate the Simplified Formula

Now, you can integrate the simplified function. The integral of \(x\) is \( \frac{1}{2}x^2\) and the integral of 1 is \(x\). Therefore the integral \( \int (x + 1) \, dx \) equals \( \frac{1}{2}x^2 + x \) . This integral can be found either manually or using the computer algebra system.
04

Add the Constant of Integration

Recall that whenever you integrate, a constant of integration must be included in the result. Thus, the final answer for the integral is \( \frac{1}{2}x^2 + x + C \).

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Most popular questions from this chapter

In Exercises 75鈥�82, find the indefinite integral using the formulas from Theorem 5.20. $$ \int \frac{1}{2 x \sqrt{1-4 x^{2}}} d x $$

In Exercises 87鈥�90, solve the differential equation. $$ \frac{d y}{d x}=\frac{1}{(x-1) \sqrt{-4 x^{2}+8 x-1}} $$

Find the derivative of the function. \(y=25 \arcsin \frac{x}{5}-x \sqrt{25-x^{2}}\)

Use a computer algebra system to find the linear approximation \(P_{1}(x)=f(a)+f^{\prime}(a)(x-a)\) and the quadratic approximation \(P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}\) of the function \(f\) at \(x=a\) . Sketch the graph of the function and its linear and quadratic approximations. \(f(x)=\arctan x, \quad a=1\)

In Exercises 106鈥�108, verify the differentiation formula. $$ \frac{d}{d x}\left[\cosh ^{-1} x\right]=\frac{1}{\sqrt{x^{2}-1}} $$
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