Mathematics

$$\int {{3^x}\cos5x\,dx = } $$


ANSWER

$$\dfrac{{{3^x}}}{{{{\left( {\log 3} \right)}^2} + 25}}\left[ {\left( {\log 3} \right).\cos5x + 5\sin5x} \right] + c$$


SOLUTION
$$I = \int {{3^x}\,\cos \,5x\,\,dx}  \cdots \left( i \right)$$

$$I = \cfrac{{\cos 5x\,{3^x}}}{{\log 3}} + \int {\cfrac{{5\sin 5x}}{{\log 3}}dx} $$

$$I = \cfrac{{{3^x}\cos 5x}}{{\log 3}} + \cfrac{5}{{\log 3}}\left[ {\cfrac{{\sin 5x\,{3^{x\,}}}}{{\log 3}} - \int {\cfrac{{5\cos 5x\,{3^x}}}{{\log 3}}} \,\,dx} \right]$$

$$I = \cfrac{{{3^x}\cos 5x}}{{\log 3}} + \cfrac{{5\sin 5\sin x\,{3^x}}}{{{{\left( {\log 3} \right)}^2}}} - \cfrac{{25}}{{{{\left( {\log 3} \right)}^2}}}$$

$$I + \cfrac{{25I}}{{{{\left( {\log 3} \right)}^2}}} = \cfrac{{{3^x}\cos 5x}}{{\log 3}} + \cfrac{{5\sin 5x\,{3^x}}}{{{{\left( {log3} \right)}^2}}}$$

$$I = \cfrac{{25 + {{\left( {\log 3} \right)}^2}}}{{{{\left( {\log 3} \right)}^2}}}\left[ {\cfrac{{{3^x}\cos 5x}}{{\log 3}} + \cfrac{{5\sin 5x\,{3^x}}}{{{{\left( {\log 3} \right)}^2}}}} \right]$$
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Single Correct Medium Published on 17th 09, 2020
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