$\int a^x \, da = $

If $\int \tan ^4 x dx = a \tan ^3 x + b \tan x + c x + k$ (where $k$ is the constant of integration),then the value of $a - b + c =$

$A$ gardener is digging a plot of land. As he gets tired,he works more slowly. After $t$ minutes,he is digging at a rate of $\frac{2}{\sqrt{t}}$ square metres per minute. How long will it take him to dig an area of $40$ square metres?

$\int {\frac{{\cot x \tan x}}{{{{\sec }^2}x - 1}}} \;dx = $

$\int \frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}} dx$ is equal to (where $c$ is a constant of integration).

Find the integral of the function $\sin ^{2}(2 x+5)$.