Mathematics

Solve $$\int {x.\sqrt[5]{{x + 1}}dx} $$


SOLUTION
$$\int x.\sqrt[5]{x+1}dx...(1)$$
put $$x+1=t^{5}….(2)$$
hence, $$x=t^{5}-  1...(3)$$  and $$dx=5t^{4} dt...(4)$$
substitute equation (2) (3) and (4) in equation (1) we get
$$\int(t^{5}-1)(t)(5t^{4})dt$$
$$\int5(t^{5}-1)t^{5}dt$$
$$5\int(t^{10}-t^{5})dt$$
$$5\bigg[\dfrac{t^{11}}{11}-\dfrac{t^{6}}{6}\bigg]$$
now, put value of $$t=(x+1)^{\dfrac{1}{5}}$$ from equation (2) in above equation we get.
$$I=5\bigg[ \dfrac{(x+1)^{\dfrac{11}{5}}}{11}-\dfrac{(x+1)^{\dfrac{6}{5}}}{6}\bigg]+C$$
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Subjective Medium Published on 17th 09, 2020
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