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Question

# Find the indefinite integral.$\int \cosh 4 x d x$

Solution

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Let $u = 4x \Rightarrow du =4dx \Rightarrow \dfrac{du}{4}=dx$. We substitute to obtain :

\begin{align*} \int \cosh (4x) dx&=\int \cosh u \dfrac{du}{4}&\color{#c34632} \text{(Integral in terms of u)}\\\\ &=\dfrac{1}{4} \int \cosh u\;du &\color{#c34632} \text{(Constant Multiple Rule)}\\\\ &=\dfrac{1}{4} \sinh u +C&\color{#c34632} \text{(Antiderivative in terms of u)} \\\\ &=\color{#4257b2}\boxed{\dfrac{1}{4} \sinh (4x)+C}&\color{#c34632} \text{(Antiderivative in terms of x)} \end{align*}

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