Question

slove this integrals

Answer 1

Answer:

For the first question:

$\int \:\frac{1}{x-x^3}dx=\int \:\frac{1}{x\left(1-x^2\right)}=\int \:\frac{1}{x\left(1+x\right)\left(1-x\right)}dx=\int \:\frac{1}{x}-\frac{1}{2x+2}-\frac{1}{2x-2}dx=\int \:\frac{1}{x}dx-\int \frac{1}{2x+2}dx-\int \frac{1}{2x-2}dx=ln\left(x\right)-\frac{1}{2}ln\left(2x+2\right)-\frac{1}{2}ln\left(2x-2\right)+C$

The key technique here is to use partial fractions, then use integration by substitution to solve integrals individually derived from partial fractions. For this particular question, you always check whether or not you can use substitution method first, then look out for polynomial fraction of the form $\frac{a}{a-x^2 }$ where 'a' is a constant because this gives you an inverse tangent antiderivative after integrating the fraction. If the two methods previously mentioned are not applicable, then go ahead using the partial fraction method, as illustrated in this question.

For the second question:

Can't read what's after x^2+1

Hope this helps :)

Step-by-step explanation: