Biology: Indefinite integral of x squared over x cubed + 1?

Indefinite integral of x squared over x cubed +

Ответ:

nataliemoore1974

17.01.2022

If you're using the app, try seeing this answer through your browser:

_______________

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

0,0(0 оценок)

Indefinite integral of x squared over x cubed + 1?

Ответ:

Ответ:

Ask your question to the AI advisor

More tips

Answers on questions: Biology

Ask an AI advisor a question