Primitive of Sine of a x over x squared

Theorem

$\ds \int \frac {\sin a x \rd x} {x^2} = -\frac {\sin a x} x + a \int \frac {\cos a x \rd x} x + C$

Proof

With a view to expressing the primitive in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

\(\ds u\)

\(=\)

\(\ds \sin a x\)

\(\ds \leadsto \ \ \)

\(\ds \frac {\d u} {\d x}\)

\(=\)

\(\ds a \cos a x\)

Derivative of $\sin a x$

and let:

\(\ds \frac {\d v} {\d x}\)

\(=\)

\(\ds \frac 1 {x^2}\)

\(\ds \leadsto \ \ \)

\(\ds v\)

\(=\)

\(\ds - \frac 1 x\)

Primitive of Power

Then:

\(\ds \int \frac {\sin a x \rd x} {x^2}\)

\(=\)

\(\ds \int \sin a x \frac 1 {x^2} \rd x\)

\(\ds \)

\(=\)

\(\ds \sin a x \paren {-\frac 1 x} - \int \paren {-\frac 1 x} a \cos a x \rd x + C\)

Integration by Parts

\(\ds \)

\(=\)

\(\ds -\frac {\sin a x} x + a \int \frac {\cos a x \rd x} x + C\)

simplifying

$\blacksquare$

Also see

• Primitive of $\dfrac {\cos a x} x$

• Primitive of $\dfrac {\cos a x} {x^2}$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$: $14.344$