What is the integral of ((1+sin(x)))/(x^3) ?

The integral of ((1+sin(x)))/(x^3) is -1/(2x^2)-(sin(x)}{2x^2}+1/2 (-\frac{cos(x))/x-\Si(x))+C

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integral of ((1+sin(x)))/(x^3)

\int \frac{( 1 + sin ( x ) )}{x ^{3}} d x

- \frac{1}{2 x ^{2}} - \frac{sin ( x )}{2 x ^{2}} + \frac{1}{2} (- \frac{cos ( x )}{x} - Si (x)) + C

Solution steps

\int \frac{1 + sin ( x )}{x ^{3}} d x

Expand

\frac{1 + sin ( x )}{x ^{3}} : \frac{1}{x ^{3}} + \frac{sin ( x )}{x ^{3}}

= \int \frac{1}{x ^{3}} + \frac{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{1}{x ^{3}} d x + \int \frac{sin ( x )}{x ^{3}} d x

\int \frac{1}{x ^{3}} d x = - \frac{1}{2 x ^{2}}

\int \frac{sin ( x )}{x ^{3}} d x = - \frac{sin ( x )}{2 x ^{2}} + \frac{1}{2} (- \frac{cos ( x )}{x} - Si (x))

= - \frac{1}{2 x ^{2}} - \frac{sin ( x )}{2 x ^{2}} + \frac{1}{2} (- \frac{cos ( x )}{x} - Si (x))

Add a constant to the solution

= - \frac{1}{2 x ^{2}} - \frac{sin ( x )}{2 x ^{2}} + \frac{1}{2} (- \frac{cos ( x )}{x} - Si (x)) + C

What is the integral of ((1+sin(x)))/(x^3) ?
The integral of ((1+sin(x)))/(x^3) is -1/(2x^2)-(sin(x)}{2x^2}+1/2 (-\frac{cos(x))/x-\Si(x))+C