Primitive of x by Cosine of x/Proof 3

Theorem

$\ds \int x \cos x \rd x = \cos x + x \sin x + C$

Proof

We have:

\(\ds \map {\dfrac \d {\d x} } {\cos x + x \sin x}\)

\(=\)

\(\ds \map {\dfrac \d {\d x} } {\cos x} + \map {\dfrac \d {\d x} } {x \sin x}\)

Sum Rule for Derivatives

\(\ds \)

\(=\)

\(\ds \map {\dfrac \d {\d x} } {\cos x} + x \map {\dfrac \d {\d x} } {\sin x} + \map {\dfrac \d {\d x} } x \sin x\)

Product Rule for Derivatives

\(\ds \)

\(=\)

\(\ds -\sin x + x \cos x + \sin x\)

Derivative of Cosine Function, Derivative of Sine Function, Derivative of Identity Function

\(\ds \)

\(=\)

\(\ds x \cos x\)

simplifying

Hence the result by definition of primitive.

$\blacksquare$

Sources

• 1945: A. Geary, H.V. Lowry and H.A. Hayden: Advanced Mathematics for Technical Students, Part I ... (previous) ... (next): Chapter $\text {III}$: Integration: Three rules for integration: $\text {I}$