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Question

Find the indefinite integral.

cosh4xdx\int \cosh 4 x d x

Solution

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Answered 9 months ago
Answered 9 months ago
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Let u=4xdu=4dxdu4=dxu = 4x \Rightarrow du =4dx \Rightarrow \dfrac{du}{4}=dx. We substitute to obtain :

cosh(4x)dx=coshudu4(Integral in terms of u)=14coshu  du(Constant Multiple Rule)=14sinhu+C(Antiderivative in terms of u)=14sinh(4x)+C(Antiderivative in terms of x)\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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