Vtyjkyuk

Solve $int_0^2014fracsqrt2014-xsqrtx+sqrt2014-xdx $.

Answer is 1007.

I tried multiplying $sqrtx-sqrt2014-x;$,

which results in $fracsqrt2014-x(sqrtx-sqrt2014-x)2x-2014=$$fracsqrt2014x-x^22x-2014-... \ =fracsqrt2014/x-12-2014/x-...
$

I got stuck so I tried substituting $u=2014-x$,

thus $int_0^2014fracusqrt2014-u+udu=... ?$

I found the value of 1007 using the value of integrand at x=0, 1007 and 2014.

But cannot solve integral. How can it be solved?

Let $I = int_0^2014fracsqrt2014-xsqrtx+sqrt2014-xdx$. Then, as you said, consider the substitution $u = 2014 - x$.

In particular,

$$I = int_2014^0 fracsqrtusqrt2014 - u + sqrtu cdot (-1) ,du = int_0^2014 fracsqrtusqrt2014 - u + sqrtu ,du$$

Relabel the $u$ as $x$, then, adding the integrals together, we get
$$2I = int_0^2014 1 ,dx$$
which gives you the result.

If you substitute $x mapsto 2014-x$, you get
$$ I = int_0^2014 fracsqrt2014-xsqrtx + sqrt2014-x , dx = int_0^2014 fracsqrtxsqrtx + sqrt2014-x , dx. $$
Hence
beginalign
1007 &= frac12int_0^2014 dx = frac12int_0^2014fracsqrt2014-xsqrtx + sqrt2014-x + fracsqrtxsqrtx + sqrt2014-x , dx\
&= frac12(I+I) = I.
endalign

$$let int_0^2014fracsqrt2014-xsqrtx+sqrt2014-xdx=A \ put 2014-x=u \ implies int_0^2014fracsqrt2014-xsqrtx+sqrt2014-xdx=int_2014^0fracsqrtusqrtu+sqrt2014-u(-du)\=int_0^2014fracsqrtusqrtu+sqrt2014-u(du)=A \ add both integrals implies 2(A)=int_0^2014dx implies A=1007$$

Don't need to do that substitution again and again just apply this formula

$$int_a^bf(x)~dx=int_a^bf(a+b-x)~dx$$

$$I=int_0^2014fracsqrt2014-xsqrtx+sqrt2014-xdx$$

Apply the formula

$$int_a^bf(x)~dx=int_a^bf(a+b-x)~dx$$

$$ I=int_0^2014fracsqrt2014-(2014+0-x)sqrt2014+0-x+sqrt2014-(2014+0-x)dx=int_0^2014fracsqrt xsqrt2014-x+sqrtxdx$$

add both integrals

$$2I=int_0^2014dx$$
$$2I=2014$$
$$I=1007$$