How to integrate sqrt(1+x^(1/3))
Published On :2021-01-27 12:48:00
[∫1/√(1+∛x) dx]
By Substitution
begin{array}{c}
text { Suppose } begin{aligned}
Rightarrow z &=sqrt[3]{x} \
z^{3} &=x
end{aligned} \
3 z^{2} d z=d x \
int frac{3 z^{2}}{sqrt{1+z}} mathrm{~d} z
end{array}
By Substitution
[Suppose⇒u=√(1+z)]
begin{array}{l} u^{2}=1+z quad, quad z^{2}=left(u^{2}-1right)^{2} \ 2 u d u=d z \ int frac{3left(u^{2}-1right)^{2}}{u} d u \ 3 int frac{u^{4}-2 u^{2}+1}{u} d u \ 3 int u^{3}-2 u+frac{1}{u} d u \ 3left(frac{u^{4}}{4}-u^{2}+ln uright)+c \ 3left(frac{(sqrt{1+z})^{4}}{4}-(sqrt{1+z})^{2}+ln |sqrt{1+z}|right)+c \ 3left(frac{(sqrt{1+sqrt[3]{x}})^{4}}{4}-(sqrt{1+sqrt[3]{x}})^{2}+ln |sqrt{1+sqrt[3]{x}}|right)+c end{array}