/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 33 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 33

Evaluate the following integrals. $$\int \frac{d x}{x^{2}-2 x+10}$$

Short Answer

Expert verified
Answer: The solution to the integral is $\frac{1}{3}\arctan\frac{x-1}{3} + C$, where C is the constant of integration.

Step by step solution

01

Identifying the appropriate substitution

From the given integral, we have the denominator as \(x^2 - 2x + 10\). We aim to represent it in the form \((x-a)^2 + b^2\). We do this by completing the square for the given quadratic denominator: \(x^2 - 2x + 10 = (x^2 - 2x + 1) + 9 = (x - 1)^2 + 3^2\) Now, we have the integral in the form: $$\int \frac{d x}{(x-1)^{2}+3^{2}}$$.
02

Performing the substitution

Now that we have the integral in the desired form, we proceed by applying the rule for the integral of \(\frac{d x}{(x-a)^{2}+b^{2}}\), where \(a=1\) and \(b=3\), as derived in step 1. Thus, the function becomes: $$\int \frac{d x}{(x-1)^{2}+3^{2}} = \frac{1}{3}\arctan\frac{x-1}{3} + C$$
03

Writing the final answer

The solution to the given integral is: $$\int \frac{d x}{x^{2}-2 x+10} = \frac{1}{3}\arctan\frac{x-1}{3} + C$$ where C is the constant of integration.

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Most popular questions from this chapter

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An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution \(u=\tan (x / 2)\) or \(x=2 \tan ^{-1} u .\) The following relations are used in making this change of variables. $$A: d x=\frac{2}{1+u^{2}} d u \quad B: \sin x=\frac{2 u}{1+u^{2}} \quad C: \cos x=\frac{1-u^{2}}{1+u^{2}}$$ $$\text { Evaluate } \int \frac{d x}{1+\sin x+\cos x}$$
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