Mathematics

Solve
$$\displaystyle \int \dfrac{dx}{(x^{1/2} + x^{1/3})} $$


SOLUTION
$$I=\displaystyle\int\dfrac{dx}{x^{\dfrac{1}{2}}+x^{\dfrac{1}{3}}}$$
Put $$x=t^{6}$$
$$dx=6t^{3}dt$$
$$=\displaystyle\int\dfrac{6t^{5}dt}{t^{3}+t^{2}}=\int\dfrac{6t^{3}}{t+1}dt$$
$$=6\displaystyle\int\left(\dfrac{t^{3}+1}{t+1}-\dfrac{1}{t+1}\right)dt=6\int \left((t^{2}-t+1)dt-\dfrac{1}{t+1}\right)dt$$
$$=6\displaystyle\int\left[(t^{2}-t+1)dt-\dfrac{1}{t+1}dt\right]$$
$$6\left[\dfrac{t^{3}}{3}-\dfrac{t^{2}}{2}+t-\ln(t+1)\right]+C$$
$$=2x^{\dfrac{1}{2}}-3x^{\dfrac{1}{3}}-6\log (x^{\dfrac{1}{6}}+1)+C$$
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Subjective Medium Published on 17th 09, 2020
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