Solve ∫ cos^4θ/1-sinθ wrt x=

$$\int \frac { \cos ^ { 4 } \theta } { 1 - \sin \theta } d x =$$

Evaluate

$$\frac{x\left(\cos(\theta )\right)^{4}}{-\sin(\theta )+1}+С$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =2\pi n_{1}+\frac{\pi }{2}$$

Solution Steps

Find the integral of \frac{\left(\cos(\theta )\right)^{4}}{1-\sin(\theta )} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{\left(\cos(\theta )\right)^{4}}{1-\sin(\theta )}x$$

Simplify.

$$\frac{\left(\cos(\theta )\right)^{4}x}{1-\sin(\theta )}$$

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

$$\begin{matrix}\frac{\left(\cos(\theta )\right)^{4}x}{1-\sin(\theta )}+С_{3},&\end{matrix}$$

Differentiate w.r.t. x

$$\frac{\left(\cos(\theta )\right)^{4}}{-\sin(\theta )+1}$$