We can solve the integral $\int\frac{x}{x^2-1}dx$ by applying integration method of trigonometric substitution using the substitution
$x=\sec\left(\theta \right)$
Intermediate steps
2
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
Expand the fraction $\frac{1+\tan\left(\theta \right)^2}{\tan\left(\theta \right)}$ into $2$ simpler fractions with common denominator $\tan\left(\theta \right)$
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
Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more
Given a function f(x) and the interval [a,b], the definite integral is equal to the area that is bounded by the graph of f(x), the x-axis and the vertical lines x=a and x=b