Blog: Find the integral of `x^3/(1+25x^2)` dx. I already know the answer but I cant find the solution step-by-step. Thank you in Advice.

Sunday, January 29, 2012

Find the integral of $\displaystyle\frac{{{x}}^{{3}}}{{{1}+{25}{{x}}^{{2}}}}$ dx. I already know the answer but I cant find the solution step-by-step. Thank you in Advice.

Hello!

Express the function under integral as $\displaystyle\frac{{{{x}}^{{2}}\cdot{x}}}{{{1}+{25}{{x}}^{{2}}}},$ observe that $\displaystyle{x}\cdot{\left.{d}{x}\right.}=\frac{{1}}{{2}}{d}{\left({{x}}^{{2}}\right)}$ and make the substitution $\displaystyle{{x}}^{{2}}={u}.$ Then $\displaystyle{d}{u}={2}{x}{\left.{d}{x}\right.}$ and the integral becomes

$\displaystyle\frac{{1}}{{2}}\int\frac{{{u}}}{{{1}+{25}{u}}}{d}{u}.$

It is simple to extract a proper part from the fraction $\displaystyle\frac{{u}}{{{1}+{25}{u}}},$

$\displaystyle\frac{{u}}{{{1}+{25}{u}}}=\frac{{1}}{{25}}{\left({1}-\frac{{1}}{{{1}+{25}{u}}}\right)}.$

So the integral is

$\displaystyle\frac{{1}}{{50}}\int{\left({1}-\frac{{1}}{{{1}+{25}{u}}}\right)}{d}{u}=\frac{{1}}{{50}}{\left({u}-\frac{{1}}{{25}}{\ln}{\left|{1}+{25}{u}\right|}\right)}+{C},$

or in terms of x

$\displaystyle\frac{{1}}{{50}}{\left({{x}}^{{2}}-\frac{{1}}{{25}}{\ln{{\left({1}+{25}{{x}}^{{2}}\right)}}}\right)}+{C}.$

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Sunday, January 29, 2012

Find the integral of $\displaystyle\frac{{{x}}^{{3}}}{{{1}+{25}{{x}}^{{2}}}}$ dx. I already know the answer but I cant find the solution step-by-step. Thank you in Advice.

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How does author Elie Wiesel use symbolism to contribute to the meaning of Night?

Sunday, January 29, 2012

Find the integral of dx. I already know the answer but I cant find the solution step-by-step. Thank you in Advice.

No comments:

Post a Comment

How does author Elie Wiesel use symbolism to contribute to the meaning of Night?

Find the integral of $\displaystyle\frac{{{x}}^{{3}}}{{{1}+{25}{{x}}^{{2}}}}$ dx. I already know the answer but I cant find the solution step-by-step. Thank you in Advice.