Solve ∫ x^4+8x^8sinx/x^8 wrt x

$$\int \frac { x ^ { 4 } + 8 x ^ { 8 } \sin x } { x ^ { 8 } } d x$$

Evaluate

$$-8\cos(x)-\frac{1}{3x^{3}}+С$$

Solution Steps

Simplify.

$$-8\cos(x)-\frac{8\cos(x)x^{2}}{3}-8x\sin(x)+\frac{\frac{1}{3}\left(-1+24\sin(x)x^{4}+3\times \frac{8\cos(x)x^{5}}{3}\right)}{x^{3}}$$

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.

$$-8\cos(x)-\frac{8\cos(x)x^{2}}{3}-8x\sin(x)+\frac{\frac{1}{3}\left(-1+24\sin(x)x^{4}+3\times \frac{8\cos(x)x^{5}}{3}\right)}{x^{3}}+С$$

Differentiate w.r.t. x

$$8\sin(x)+\frac{1}{x^{4}}$$