What is the integral of (x^2cos(x)-2xsin(x))/(x^4) ?

The integral of (x^2cos(x)-2xsin(x))/(x^4) is (sin(x))/(x^2)+C

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integral of (x^2cos(x)-2xsin(x))/(x^4)

\int \frac{x ^{2} cos ( x ) - 2 x sin ( x )}{x ^{4}} d x

\frac{sin ( x )}{x ^{2}} + C

Solution steps

\int \frac{x ^{2} cos ( x ) - 2 x sin ( x )}{x ^{4}} d x

Expand

\frac{x ^{2} cos ( x ) - 2 x sin ( x )}{x ^{4}} : \frac{cos ( x )}{x ^{2}} - \frac{2 sin ( x )}{x ^{3}}

= \int \frac{cos ( x )}{x ^{2}} - \frac{2 sin ( x )}{x ^{3}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{cos ( x )}{x ^{2}} d x - \int \frac{2 sin ( x )}{x ^{3}} d x

\int \frac{cos ( x )}{x ^{2}} d x = - \frac{cos ( x )}{x} - Si (x)

\int \frac{2 sin ( x )}{x ^{3}} d x = - \frac{sin ( x )}{x ^{2}} - \frac{cos ( x )}{x} - Si (x)

= - \frac{cos ( x )}{x} - Si (x) - (- \frac{sin ( x )}{x ^{2}} - \frac{cos ( x )}{x} - Si (x))

Simplify

= \frac{sin ( x )}{x ^{2}}

Add a constant to the solution

= \frac{sin ( x )}{x ^{2}} + C

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What is the integral of (x^2cos(x)-2xsin(x))/(x^4) ?
The integral of (x^2cos(x)-2xsin(x))/(x^4) is (sin(x))/(x^2)+C