# Thread: How to solve this integral using substitution?

1. ## How to solve this integral using substitution?

$$int{(x-(\sqrt{x}-5)^2)^2} dx$$
I find the answer using expansion. How to solve it using substitution?

2. ## Re: How to solve this integral using substitution?

To me an obvious first step is to let $u= \sqrt{x}= x^{1/2}$. Then $x= u^2$ and $dx= 2udu$.

The integral becomes $2\int [u^2- (u- 5)^2](u du)= 2\int (10u- 25)udu= 10\int 2u^2- 5u du$.

3. ## Re: How to solve this integral using substitution?

It will be $$\frac{20}{3}u^3 - \frac{50}{2}u^2$$
But it still wrong..

4. ## Re: How to solve this integral using substitution?

Originally Posted by Helly123
It will be $$\frac{20}{3}u^3 - \frac{50}{2}u^2$$
But it still wrong..
Go ahead and use $u=\sqrt{x}$.

5. ## Re: How to solve this integral using substitution?

Originally Posted by Helly123
It will be $$\frac{20}{3}u^3 - \frac{50}{2}u^2$$
But it still wrong..
That is because of a small typo that HallsofIvy made. His method is spot on.

\displaystyle \begin{align*}\int \big( x- (\sqrt{x}-5)^2 \big)^2 dx & = 2\int \big(u^2-(u-5)^2 \big)^2(udu) \\ & = 2\int (10u-25)^2udu \\ & = 50\int (4u^3-20u^2+25u)du \\ & = 50\left(u^4 - \dfrac{20}{3}u^3+\dfrac{25}{2}u^2\right) + C \\ & = 50x^2-\dfrac{1000}{3}x^{3/2} + 625x + C\end{align*}

Halls just missed an exponent of 2.

6. ## Re: How to solve this integral using substitution?

Ok thank you so much