Question

Indefinite integral of x squared over x cubed + 1?

Answer 1

If you're using the app, try seeing this answer through your browser:  brainly.com/question/2862768

_______________


Evaluate the indefinite integral:

$\mathsf{\displaystyle\int\! \frac{x^2}{x^3+1}\,dx}\\\\\\ \mathsf{=\displaystyle\int\! \frac{1}{3}\cdot 3\cdot \frac{x^2}{x^3+1}\,dx}\\\\\\ \mathsf{=\displaystyle \frac{1}{3}\int\! \frac{1}{x^3+1}\cdot 3x^2\,dx\qquad\quad(i)}$


Make a substitution:

$\mathsf{x^3+1=u\quad \Rightarrow\quad 3x^2\,dx=du}$


and the integral (i) becomes

$\mathsf{=\displaystyle \frac{1}{3}\int\! \frac{1}{u}\,du}\\\\\\ \mathsf{=\displaystyle \frac{1}{3}\cdot \ell n\!\left|u\right|+C}$


Substitute back for u = x³ + 1, and you get the result:

$\mathsf{=\dfrac{1}{3}\,\ell n\!\left|x^3+1\right|+C}$


$\therefore~~\mathsf{\displaystyle\int\!\frac{x^2}{x^3+1}\,dx=\frac{1}{3}\,\ell n\!\left|x^3+1\right|+C}\qquad\quad\checkmark$


I hope this helps. =)


Tags:  indefinite integral rational function substitution natural logarithm log ln differential integral calculus