Find the antiderivative of sqrt(5 + sqrtx)?

Question

Find ∫√(5 + √x)dx

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Answer 1

Use substitution. Let u=5+√x. Then du=1/(2√x) dx, and dx=2 √x du. Substitute this into the original equation to get:

∫√u 2√x du

But note that we set u=5+√x. Thus, √x=u-5. So we have:

2∫√u (u-5) du

Which becomes:
2∫u^(3/2) - 5u^(1/2) du

Which is easily integrated to yield:

4u^(5/2)/5 - 20u^(3/2)/3 + C

Substituting u back into the equation gives our final answer:

4(5+√x)^(5/2)/5 - 20(5+√x)^(3/2)/3 + C

Answer 2

let √(5 + √x) = y
Therefore,
x = y^4 - 10y^2 + 25
and thus,
dx = 4y(y^2 - 5)dy
substituting these values in the starting integral our problem changes to,
4∫(y^4 - 5y^2)dy
which can be simply solved to give the result

(4y^5)/5 - (20y^3)/3

putting back the value of y what we have initially assumed we will get our result in 'x' form

Answer 3

I dont think neone would be able to do this by hand, but Mathematica says it's: 4/15 * (5+Sqrt(x))^3/2 * (-10+3Sqrt(x))

Answer 4

(1/3sqrtx)(5+sqrtx)to the power3/2

Answer 5

put sqrtx as 5cost and try if you still dont get it ask

Answer 6

ANSWER IS X√(5 + √X)