Calculating Integral of \int \frac{dx}{x(x^4+1)} using Substitution

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In summary, solving integrals using substitution involves identifying a part of the integral that can be substituted with a new variable and choosing a suitable substitution. The general formula for substitution in integration is ∫f(g(x))g'(x)dx = ∫f(u)du, where u = g(x) and du = g'(x)dx. To choose a suitable substitution, one can look for a part of the integral that resembles a derivative of another function or try substituting with different variables. The substitution used for calculating the integral of \int \frac{dx}{x(x^4+1)} is u = x^4+1, which simplifies the integral to \int \frac{1}{u}du.
  • #1
nameVoid
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[tex]
\int \frac{dx}{x(x^4+1)}
[/tex]
[tex]
u=x^2
[/tex]
[tex]
\sqrt{u}=x
[/tex]
[tex]
dx=\frac{1}{2\sqrt{u}}
[/tex]
[tex]
\frac{1}{2}\int \frac{du}{u^2+1}
[/tex]
[tex]
\frac{1}{2}arctanx^2+C
[/tex]
 
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  • #2
nameVoid said:
[tex]
\int \frac{dx}{x(x^4+1)}
[/tex]
[tex]
u=x^2
[/tex]
[tex]
\sqrt{u}=x
[/tex]
[tex]
dx=\frac{1}{2\sqrt{u}}
[/tex]
[tex]
\frac{1}{2}\int \frac{du}{u^2+1}
[/tex]
The integral above isn't right. You forgot to replace the x factor in the denominator.
nameVoid said:
[tex]
\frac{1}{2}arctanx^2+C
[/tex]
 
  • #3
[tex]
\frac{1}{2} \int \frac{du}{u(u^2+1)}=\int \frac{A}{u}+\frac{Bu+C}{u^2+1}du
[/tex]
[tex]
A=\frac{1}{2}=-B
[/tex]
[tex]
ln|x|-\frac{1}{4}ln(x^4+1)+C
[/tex]
 

1. How do you approach solving integrals using substitution?

To solve integrals using substitution, you must first identify a part of the integral that can be substituted with a new variable. Then, you must choose a suitable substitution that will simplify the integral and make it easier to solve. After substituting, you can solve the new integral using basic integration rules.

2. What is the general formula for substitution in integration?

The general formula for substitution in integration is ∫f(g(x))g'(x)dx = ∫f(u)du, where u = g(x) and du = g'(x)dx.

3. How do you choose a suitable substitution for a given integral?

You can choose a suitable substitution by looking for a part of the integral that resembles a derivative of another function. This can be done by recognizing common derivative patterns or using trigonometric identities. You can also try substituting with different variables until you find one that simplifies the integral.

4. What is the substitution used for calculating the integral of \int \frac{dx}{x(x^4+1)}?

The substitution used for this integral is u = x^4+1. This will simplify the integral to \int \frac{1}{u}du, which can be easily solved using basic integration rules.

5. Can you provide a step-by-step explanation for solving the integral of \int \frac{dx}{x(x^4+1)} using substitution?

Sure, first we substitute u = x^4+1, so du = 4x^3dx. We can then rewrite the integral as \int \frac{1}{u}du. Next, we solve the new integral, which is ln|u| + C. Substituting back u = x^4+1, we get ln|x^4+1| + C as the final answer.

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