Solve ∫ x=acos^3t

$$\int x = a \cos ^ { 3 } t$$

Solve for a

$$\left\{\begin{matrix}a=\frac{x^{2}+2С}{2\left(\cos(t)\right)^{3}}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }t=\pi n_{1}+\frac{\pi }{2}\\a\in \mathrm{R}\text{, }&С=-\frac{x^{2}}{2}\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }t=\pi n_{1}+\frac{\pi }{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$a\left(\cos(t)\right)^{3}=\int x\mathrm{d}x$$

The equation is in standard form.

$$\left(\cos(t)\right)^{3}a=\frac{x^{2}}{2}+С$$

Divide both sides by \left(\cos(t)\right)^{3}.

$$\frac{\left(\cos(t)\right)^{3}a}{\left(\cos(t)\right)^{3}}=\frac{\frac{x^{2}}{2}+С}{\left(\cos(t)\right)^{3}}$$

Dividing by \left(\cos(t)\right)^{3} undoes the multiplication by \left(\cos(t)\right)^{3}.

$$a=\frac{\frac{x^{2}}{2}+С}{\left(\cos(t)\right)^{3}}$$

Divide \frac{x^{2}}{2}+С by \left(\cos(t)\right)^{3}.

$$a=\frac{x^{2}+2С}{2\left(\cos(t)\right)^{3}}$$