Theorem: Linear Combination of Complex Integrals

Let $\closedint a b$ be a closed real interval.
Let $f, g: \closedint a b \to \C$ be complex Riemann integrable functions over $\closedint a b$.
Let $\lambda, \mu \in \C$ be complex constants.
Then:
:$\ds \int_a^b \paren {\lambda \map f t + \mu \map g t} \rd t = \lambda \int_a^b \map f t \rd t + \mu \int_a^b \map g t \rd t$

Proof:
First, we prove the result for addition only without multiplying by $\lambda, \mu$:
{{begin-eqn}}
{{eqn | l = \int_a^b \map f t + \map g t \rd t
      | r = \int_a^b \map \Re {\map f t + \map g t} \rd t + i \int_a^b \map \Im {\map f t + \map g t} \rd t
      | c = {{Defof|Complex Riemann Integral}}
}}
{{eqn | r = \int_a^b \map \Re {\map f t} + \map \Re {\map g t} \rd t
}}
{{eqn | o =
      | ro= +
      | r = i \int_a^b \map \Im {\map f t} + \map \Im {\map g t} \rd t
      | c = Addition of Real and Imaginary Parts
}}
{{eqn | r = \int_a^b \map \Re {\map f t} \rd t + \int_a^b \map \Re {\map g t} \rd t
}} 
{{eqn | o =
      | ro= +
      | r = i \paren {\int_a^b \map \Im {\map f t} \rd t + \int_a^b \map \Im {\map g t} \rd t}
      | c = Linear Combination of Definite Integrals
}}
{{eqn | r = \int_a^b \map f t \rd t + \int_a^b \map g t \rd t
}}
{{end-eqn}}
{{qed|lemma}}
Next, we show the result for only one complex integral multiplied by a constant $\lambda$.
By definition of complex number, we can find $\lambda_x, \lambda_y \in \R$ so $\lambda = \lambda_x + i \lambda_y$.
Then:
{{begin-eqn}}
{{eqn | l = \int_a^b \lambda \map f t \rd t
      | r = \int_a^b \map \Re {\lambda \map f t} \rd t + i \int_a^b \map \Im {\lambda \map f t} \rd t
      | c = {{Defof|Complex Riemann Integral}}
}}
{{eqn | r = \int_a^b \map \Re {\paren {\lambda_x + i \lambda_y} \map f t} \rd t
}}
{{eqn | o =
      | ro= +
      | r = i \int_a^b \map \Im {\paren {\lambda_x + i \lambda_y} \map f t} \rd t
}}
{{eqn | r = \int_a^b \map \Re {\lambda_x \map f t} + \map \Re {i \lambda_y \map f t} \rd t
}}
{{eqn | o =
      | ro= +
      | r = i \int_a^b \map \Im {\lambda_x \map f t} + \map \Im {i \lambda_y \map f t} \rd t
      | c = Addition of Real and Imaginary Parts
}}
{{eqn | r = \int_a^b \lambda_x \map \Re {\map f t} - \lambda_y \map \Im {\map f t} \rd t
}}
{{eqn | o =
      | ro= +
      | r = i \int_a^b \lambda_x \map \Im {\map f t} + \lambda_y \map \Re {\map f t} \rd t
      | c = Multiplication of Real and Imaginary Parts
}}
{{eqn | r = \lambda_x \int_a^b \map \Re {\map f t} \rd t - \lambda_y \int_a^b \map \Im {\map f t} \rd t
}}
{{eqn | o =
      | ro= +
      | r = i \lambda_x \int_a^b \map \Im {\map f t} \rd t + i \lambda_y \int_a^b \map \Re {\map f t} \rd t
      | c = Linear Combination of Definite Integrals
}}
{{eqn | r = \lambda_x \paren {\int_a^b \map \Re {\map f t} \rd t + i \int_a^b \map \Im {\map f t} \rd t}
}}
{{eqn | o =
      | ro= +
      | r = i \lambda_y \paren {i \int_a^b \map \Im {\map f t} \rd t + \int_a^b \map \Re {\map f t} \rd t}
}}
{{eqn | r = \lambda_x \int_a^b \map f t \rd t + i \lambda_y \int_a^b \map f t \rd t
      | c = {{Defof|Complex Riemann Integral}}
}}
{{eqn | r = \lambda \int_a^b \map f t \rd t
}}
{{end-eqn}}
{{qed|lemma}}
It follows from the results above that:
{{begin-eqn}}
{{eqn | l = \lambda \int_a^b \map f t \rd t + \mu \int_a^b \map g t \rd t
      | r = \int_a^b \lambda \map f t \rd t + \int_a^b \mu \map g t \rd t
}}
{{eqn | r = \int_a^b \paren {\lambda \map f t + \mu \map g t} \rd t
}}
{{end-eqn}}
{{qed}}