Evaluating $\int_{-5}^{\sqrt{x}}(\frac{\cos t}{t^{10}})dt$ 
Evaluate $y=\int_{-5}^{\sqrt{x}}(\frac{\cos t}{t^{10}})dt$


I've tried differentiating both sides of the fraction until the denominator was 1, and then integrating that by parts, but this was marked wrong. I know I can't just integrate by parts right off the bat because differentiating $t^{-b}$ will always end in another varied reciprocal, it'll just make things worse and worse.
How can I solve this?

It is possible to find or verify the solution to the indefinite integral 
$I = {\displaystyle\int}\dfrac{\cos\left(t\right)}{t^{10}}\,dt$
by consulting or using the Integral Calculator site.
The integration solution involves several integration by parts of the known form
$\displaystyle {\int}f g' = f g - {\int}f'g$
Integrating by parts and taking
$f = \cos(t); g'=\dfrac{1}{t^{10}} \\ f' = -\sin(t); g=-\dfrac{1}{9t^9},$
we get
$I = -\dfrac{\cos\left(t\right)}{9t^9}-{\displaystyle\int}\dfrac{\sin\left(t\right)}{9t^9}\,dt \quad (1)$
Applying integration by parts to the indefinite integral in (1):
$f = \sin(t); g'= \dfrac{1}{t^9} \\ f' = \cos(t); g = -\dfrac{1}{8t^8},$
we obtain
${\displaystyle\int}\dfrac{\sin\left(t\right)}{t^9}\,dt= -\dfrac{\sin\left(t\right)}{8t^8}-{\displaystyle\int}-\dfrac{\cos\left(t\right)}{8t^8}\,dt$
Applying integration by parts in a similar way seven more times, we arrive to the integral
${\displaystyle\int}\dfrac{\sin\left(t\right)}{t}\,dt,$
which represents a definition of the sine integral $\operatorname{Si}\left(t\right).$
Adding the integration results, calculating and developing, we obtain the following solution to the indefinite integral:
$\displaystyle I = -\frac{t^9 \text{Si}(t)+\left(t^6-6 t^4+120 t^2-5040\right) t \sin (t)+\left(t^8-2 t^6+24 t^4-720 t^2+40320\right) \cos (t)}{362880 t^9} +C$
Applying the given limits or bounds of integration, we obtain the solution to the definite integral (verified with Mathematica):
$\displaystyle \int_{-5}^{\sqrt{x}} \frac{\cos (t)}{t^{10}} \, dt =\\ \displaystyle \left(\frac{1}{362880 x^{9/2}}\right)\left(x^{9/2} \left(-\text{Si}\left(\sqrt{x}\right)\right)-\sqrt{x} \left(x^3-6 x^2+120 x-5040\right) \sin \left(\sqrt{x}\right)\\ \displaystyle -\left(x^4-2 x^3+24 x^2-720 x+40320\right) \cos \left(\sqrt{x}\right)\right) \\ \displaystyle+\frac{-390625 \text{Si}(5)-9835 \sin (5)-79339 \cos (5)}{141750000000}$
It can be verified with Mathematica that the indefinite integral has the following more "compact" solution:
$\displaystyle I = -\frac{E_{10}(-i t)+E_{10}(i t)}{2 t^9}+C,$
where $E_n(z)$ is the exponential integral function.
And the given definite integral has the following solution:
$\displaystyle \int_{-5}^{\sqrt{x}} \frac{\cos (t)}{t^{10}} \, dt =\frac{-E_{10}(-5 i)-E_{10}(5 i)}{3906250}-\frac{E_{10}\left(-i \sqrt{x}\right)+E_{10}\left(i \sqrt{x}\right)}{2 x^{9/2}}$
The graphical representation below shows the real part (in blue) and the imaginary part (in red) of the last solution above to the definite integral (made with Mathematica):

