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(Solved): - where not met in 200 iterations Recall the integration by parts formuala: $\text { int } f(x) ... - where not met in 200 iterations Recall the integration by parts formuala: \[ \text { int } f(x) g^{\prime}(x) d x=f(x) g(x) \text { - int } f^{\prime}(x) g(x) d x \text { ' }$ Complete the following computation 'int $$\mathrm{e}^{\wedge}(5 \mathrm{x}) \cos (5 \mathrm{x}) \mathrm{dx}$$ ' via integration by parts. Set $' f(x)=e^{\wedge}(5 x)^{\prime} \text { and } \quad{ }^{\prime} g^{\prime}(x)=\cos (5 x)^{\prime}$ 1. Find ' $$f^{\prime}(x)^{\prime}$$ and ' $$g(x)^{\prime}$$ where ' $$g(x)^{\prime}$$ has no constant term: $\begin{array}{l} ' f^{\prime}(x)=' \\ ' g(x)=' \end{array}$ 2. Using the integration by parts formula, $\begin{array}{c} \text { int } \mathrm{e}^{\wedge}(5 x) \cos (5 x) d x{ }^{\prime}={ }^{\prime} \quad F(x)-\text { int } G(x) d x ' \\ ' F(x)=' \\ ' G(x)=' \end{array}$ 3. Using integration by parts to compute 'int $$G(x) d x^{\prime}$$, find ' $$H(x)^{\prime}$$ and a constant ' $$c$$ ' such that 'int $$\mathrm{e}^{\wedge}(5 \mathrm{x}) \cos (5 \mathrm{x}) \mathrm{dx}$$ ' $$={ }^{\prime} \quad ` \mathrm{~F}(\mathrm{x})-\left(\mathrm{H}(\mathrm{x})-\mathrm{c} \text { int } \mathrm{e}^{\wedge}(5 \mathrm{x}) \cos (5 \mathrm{x}) \mathrm{dx}\right)^{\prime}$$ $\text { ' } \mathrm{H}(\mathrm{x})=\text { ' }$ $\text { ' } \mathrm{C}=\text { ' }$ 4. Use (1), (2), (3) to compute 'int $$\mathrm{e}^{\wedge}(5 \mathrm{x}) \cos (5 \mathrm{x}) \mathrm{dx}=$$ '

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