What is the integral of (sin(x))/(pi-x) ?

The integral of (sin(x))/(pi-x) is \Si(-pi+x)+C

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integral of (sin(x))/(pi-x)

\int \frac{sin ( x )}{π - x} d x

Si (- π + x) + C

Solution steps

\int \frac{sin ( x )}{π - x} d x

Factor

π - x : - (- π + x)

= \int \frac{sin ( x )}{- ( - π + x )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{sin ( x )}{- π + x} d x

Apply u-substitution

= - \int \frac{sin ( u + π )}{u} d u

Rewrite using trig identities

= - \int - \frac{sin ( u )}{u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - (- \int \frac{sin ( u )}{u} d u)

Use the nonelementary integral:

\int \frac{sin ( x )}{x} d x = Si (x)

= - (- Si (u))

Substitute back

u = - π + x

= - (- Si (- π + x))

Simplify

= Si (- π + x)

Add a constant to the solution

= Si (- π + x) + C

What is the integral of (sin(x))/(pi-x) ?
The integral of (sin(x))/(pi-x) is \Si(-pi+x)+C