Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- Integrate by trigonometric substitution
- Integrate by partial fractions
- Integrate by substitution
- Integrate by parts
- Integrate using tabular integration
- Weierstrass Substitution
- Integrate using trigonometric identities
- Integrate using basic integrals
- Product of Binomials with Common Term
- FOIL Method
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We can solve the integral $\int\frac{x}{x^2-1}dx$ by applying integration method of trigonometric substitution using the substitution
Now, in order to rewrite $d\theta$ in terms of $dx$, we need to find the derivative of $x$. We need to calculate $dx$, we can do that by deriving the equation above
Substituting in the original integral, we get
Apply the trigonometric identity: $\sec\left(\theta \right)^2-1$$=\tan\left(\theta \right)^2$, where $x=\theta $
Simplify the fraction by $\tan\left(\theta \right)$
Applying the trigonometric identity: $\sec\left(\theta \right)^2 = 1+\tan\left(\theta \right)^2$
Expand the fraction $\frac{1+\tan\left(\theta \right)^2}{\tan\left(\theta \right)}$ into $2$ simpler fractions with common denominator $\tan\left(\theta \right)$
Simplify the resulting fractions
Expand the integral $\int\left(\frac{1}{\tan\left(\theta \right)}+\tan\left(\theta \right)\right)d\theta$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
The integral $\int\frac{1}{\tan\left(\theta \right)}d\theta$ results in: $\ln\left(\frac{\sqrt{x^2-1}}{x}\right)$
The integral $\int\tan\left(\theta \right)d\theta$ results in: $\ln\left(x\right)$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$