Solve ∫ intintfrac{xsqrt{t}}{9^2 wrt xdtdq

$$\int \int \int \frac { x \sqrt { t } } { 9 ^ { 2 } } d x d t d q$$

Evaluate

$$\frac{qt^{\frac{3}{2}}x^{2}}{243}+С\left(qt+q\right)+С_{1}$$

Solution Steps

Find the integral of \int \int \frac{x\sqrt{t}}{9^{2}}\mathrm{d}x\mathrm{d}t using the table of common integrals rule \int a\mathrm{d}q=aq.

$$\int \int \frac{x\sqrt{t}}{9^{2}}\mathrm{d}x\mathrm{d}tq$$

Simplify.

$$\left(\frac{x^{2}t^{\frac{3}{2}}}{243}+Сt+С\right)q$$

If F\left(q\right) is an antiderivative of f\left(q\right), then the set of all antiderivatives of f\left(q\right) is given by F\left(q\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\left(\frac{x^{2}t^{\frac{3}{2}}}{243}+Сt+С\right)q+С$$

Differentiate w.r.t. x

$$\frac{2qxt^{\frac{3}{2}}}{243}$$