Primitive of Exponential of a x by Hyperbolic Cosine of b x/Exponential Form

Theorem

$\ds \int e^{a x} \cosh b x \rd x = \frac {e^{a x} } 2 \paren {\frac {e^{b x} } {a + b} + \frac {e^{-b x} } {a - b} } + C$

for $a^2 \ne b^2$.

Proof

\(\ds \int e^{a x} \cosh b x \rd x\)

\(=\)

\(\ds \int e^{a x} \paren {\frac {e^{b x} + e^{-b x} } 2} \rd x\)

Definition of Hyperbolic Cosine

\(\ds \)

\(=\)

\(\ds \frac 1 2 \int e^{a x} \paren {e^{b x} + e^{-b x} } \rd x\)

Primitive of Constant Multiple of Function

\(\ds \)

\(=\)

\(\ds \frac 1 2 \int \paren {e^{a x} e^{b x} + e^{a x} e^{-b x} } \rd x\)

\(\ds \)

\(=\)

\(\ds \frac 1 2 \int \paren {e^{a x + b x} + e^{a x - b x} } \rd x\)

Product of Powers

\(\ds \)

\(=\)

\(\ds \frac 1 2 \int e^{a x + b x} \rd x + \frac 1 2 \int e^{a x - b x} \rd x\)

Linear Combination of Primitives

\(\ds \)

\(=\)

\(\ds \frac 1 2 \int e^{\paren {a + b} x} \rd x + \frac 1 2 \int e^{\paren {a - b} x} \rd x\)

\(\ds \)

\(=\)

\(\ds \frac 1 2 \frac {e^{\paren {a + b} x} } {a + b} + \frac 1 2 \frac {e^{\paren {a - b} x} } {a - b} + C\)

Primitive of $e^{a x}$

\(\ds \)

\(=\)

\(\ds \frac 1 2 \frac {e^{a x + b x} } {a + b} + \frac 1 2 \frac {e^{a x - b x} } {a - b} + C\)

\(\ds \)

\(=\)

\(\ds \frac 1 2 \frac {e^{a x} e^{b x} } {a + b} + \frac 1 2 \frac {e^{a x} e^{-b x} } {a - b} + C\)

Product of Powers

\(\ds \)

\(=\)

\(\ds \frac {e^{a x} } 2 \paren {\frac {e^{b x} } {a + b} + \frac {e^{-b x} } {a - b} } + C\)

$\blacksquare$

Also see

• Primitive of $e^{a x} \sinh b x$

Sources

• 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $138$.