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Chapter 1: Problem 7

Evaluate the following integrals : $$\int \frac{d x}{x^{4}+18 x^{2}+81}$$

Short Answer

Expert verified
Answer: The integral of the given function is $$\frac{1}{2} \ln\left|x^2+9\right| + C$$, where $$C$$ is the constant of integration.

Step by step solution

01

Factor the denominator

First, we try to manipulate the denominator to make it look like the sum of squares. We can rewrite the denominator as follows: $$x^4 + 18x^2 + 81 = \left(x^4 + 2 \cdot 3 \cdot 3x^2 + 3^4\right) = \left(x^2 + 9\right)^2$$ Now the denominator is a perfect square, which is what we wanted.
02

Substitute variables for integration

Let \(u = x^2 + 9\). Then, we will take the derivative, which will give us the following: $$\frac{du}{2} = x\, dx \Rightarrow du = 2x\, dx$$ If we substitute \(x^2 = u - 9\) into the integral, we have: $$\int \frac{1}{(u - 9) + 9}\,dx = \int \frac{1}{u}\,dx$$ But we still need to substitute \(du = 2x\,dx\) to get rid of \(x\). The equation becomes: $$\int \frac{1}{u} \cdot \frac{du}{2x}$$ Now, we cancel out \(x\) in the numerator and the denominator, and we are left with: $$\frac{1}{2} \int \frac{1}{u}\, du$$
03

Integrate and substitute back

Now, we simply need to evaluate the integral: $$\frac{1}{2} \int \frac{1}{u}\, du = \frac{1}{2} \ln\left| u \right| + C$$ Now, substitute \(u\) back in terms of \(x\): $$\frac{1}{2} \ln\left|x^2+9\right| + C$$ Finally, the integral \(\int \frac{1}{x^4+18x^2+81}\,dx\) evaluates to: $$\frac{1}{2} \ln\left|x^2+9\right| + C$$

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Most popular questions from this chapter

Evaluate the following integrals: (i) \(\int \frac{\sqrt{2 x+1}}{x^{2}} d x\) (ii) \(\int \frac{x d x}{(a+b x)^{1 / 2}}\) (iii) \(\int \sqrt{\frac{x+a}{x+b}} d x\)

Only one of the functions \(\sqrt{x} \sqrt[3]{1-x}\) and \(\sqrt{1-x} \sqrt[3]{1-x}\) has an elementary antiderivative. Find the function.

Two of these antiderivatives are elementary functions; find them. (A) \(\int \ln x d x\) (B) \(\int \frac{\ln x d x}{x}\) (C) \(\int \frac{d x}{\ln x}\)

Evaluate the following integrals: (i) \(\int \frac{5 \cos ^{3} x+3 \sin ^{3} x}{\sin ^{2} x \cos ^{2} x} d x\) (ii) \(\int\left(\cos ^{6} x+\sin ^{6} x\right) d x\) (iii) \(\int \sin ^{3} x \cos \frac{x}{2} d x\) (iv) \(\int \frac{d x}{\sqrt{3} \cos x+\sin x}\)

Evaluate the following integrals: (i) \(\int \frac{\sqrt{x}+\sqrt[3]{x}}{\sqrt[4]{x^{5}}-\sqrt[6]{x^{7}}} d x\) (ii) \(\int \frac{x^{-2 / 3}}{2 x^{1 / 3}+(x-1)^{1 / 3}} d x\) (iii) \(\int \frac{d x}{x\left(2+\sqrt[3]{\frac{x-1}{x}}\right)}\)
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