Question #300590

Evaluate this integral x(x²+4)dx / x⁴+9

1
Expert's answer
2022-02-22T05:21:55-0500
x(x2+4)x4+9dx\int \dfrac{x(x^2+4)}{x^4+9}dx

Substitute x2=u,2xdx=dux^2=u, 2xdx=du


x(x2+4)x4+9dx=12u+4u2+9du\int \dfrac{x(x^2+4)}{x^4+9}dx=\dfrac{1}{2}\int \dfrac{u+4}{u^2+9}du

=12uu2+9du+2duu2+9=\dfrac{1}{2}\int \dfrac{u}{u^2+9}du+2\int \dfrac{du}{u^2+9}

12uu2+9du\dfrac{1}{2}\int \dfrac{u}{u^2+9}du

t=u2+9,dt=2udut=u^2+9, dt=2udu

12uu2+9du=14dtt=\dfrac{1}{2}\int \dfrac{u}{u^2+9}du=\dfrac{1}{4}\int \dfrac{dt}{t}=

=14lnt+C1=14ln(u2+9)+C1=\dfrac{1}{4}\ln |t|+C_1=\dfrac{1}{4}\ln(u^2+9)+C_1

=14ln(x4+9)+C1=\dfrac{1}{4}\ln(x^4+9)+C_1

2duu2+92\int \dfrac{du}{u^2+9}

s=u3,ds=13dus=\dfrac{u}{3}, ds=\dfrac{1}{3}du

2duu2+9=23ds9s2+9=23ds1+s22\int \dfrac{du}{u^2+9}=2\int \dfrac{3ds}{9s^2+9}=\dfrac{2}{3}\int \dfrac{ds}{1+s^2}

=23tan1(s)+C2=23tan1(u3)+C2=\dfrac{2}{3}\tan^{-1}(s)+C_2=\dfrac{2}{3}\tan^{-1}(\dfrac{u}{3})+C_2


=23tan1(x23)+C2=\dfrac{2}{3}\tan^{-1}(\dfrac{x^2}{3})+C_2

Then


x(x2+4)x4+9dx=14ln(x4+9)+23tan1(x23)+C\int \dfrac{x(x^2+4)}{x^4+9}dx=\dfrac{1}{4}\ln(x^4+9)+\dfrac{2}{3}\tan^{-1}(\dfrac{x^2}{3})+C


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