Primitive of Exponential of a x by Cosine of b x/Proof 3

Proof
Let $a, b \in \R_{>0}$ be real constants.

Let $f_1$ and $f_2$ be the real functions defined as:

Let $\map \CC \R$ denote the space of continuous real-valued functions.

Let $\struct {\map {\CC^1} \R, +, \, \cdot \,}_\R$ denote the vector space of continuously differentiable real-valued functions.

Let $S = \span \set {f_1, f_2} \subset \map {\CC^1} \R$ be a vector space.

Let $D : S \to S$ be the derivative $x$.

From Differentiation of Exponential of a x by Cosine of b x and Exponential of a x by Sine of b x wrt x as Invertible Matrix, $D$ is expressible as:


 * $\mathbf D = \begin{bmatrix}

a & b \\ -b & a \end{bmatrix}$

and is invertible.

By Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant:


 * $\mathbf D^{-1} = \dfrac 1 {a^2 + b^2} \begin{bmatrix}

a & -b \\ b & a \end{bmatrix}$

Then:

Application of $\mathbf D$ on both sides on the left and writing out explicitly in terms of $f_1$ and $f_2$ yields:


 * $f_1 = \dfrac \d {\d x} \dfrac {a f_1 + b f_2} {a^2 + b^2}$

Integrating $x$:


 * $\ds \int f_1 \rd x = \frac {a f_1 + b f_2} {a^2 + b^2} + C$

where $C$ is an arbitrary constant.

Substitute definitions of $f_1$ and $f_2$ to get the desired result.