[tex]\large\underline{\sf{Solution-}}[/tex]
[tex]\sf \: \displaystyle\int\sf ( {x}^{a} + {a}^{x} + {e}^{x} {a}^{x} + sin \: ax) \: dx \\ \\ [/tex]
[tex]\sf \: = \: \displaystyle\int\sf {x}^{a}dx + \displaystyle\int\sf {a}^{x} dx+ \displaystyle\int\sf {(ea)}^{x} dx+ \displaystyle\int\sf sin \: ax \: dx \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{{x}^{a + 1}}{a + 1} + \dfrac{{a}^{x}}{loga} + \dfrac{{(ae)}^{x}}{log(ae)} - \dfrac{cos \: ax}{a} + c \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Formulae Used :-
[tex]\boxed{ \sf{ \:\displaystyle\int\sf {x}^{n} \: dx \: = \: \frac{ {x}^{n + 1} }{n + 1} + c \: }} \\ \\ [/tex]
[tex]\boxed{ \sf{ \:\displaystyle\int\sf {a}^{x} \: dx \: = \: \frac{{a}^{x}}{loga} + c\: }} \\ \\ [/tex]
[tex]\boxed{ \sf{ \:\displaystyle\int\sf {e}^{x} \: dx \: = \: {e}^{x} + c \: }} \\ \\ [/tex]
[tex]\boxed{ \sf{ \:\displaystyle\int\sf sin(ax + b) \: = \: - \: \dfrac{cos(ax + b)}{a} \: + \: c \: }} \\ \\ [/tex]
[tex] {{ \mathfrak{Additional\:Information}}}[/tex]
[tex]\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}[/tex]
Step-by-step explanation:
[tex]\large\underline{\bf{Solution}} [/tex]
[tex]\begin{gathered}\sf \: \displaystyle\int\sf ( {x}^{a} + {a}^{x} + {e}^{x} {a}^{x} + sin \: ax) \: dx \\ \\ \end{gathered} [/tex]
[tex]\begin{gathered}\sf \: = \: \displaystyle\int\sf {x}^{a}dx + \displaystyle\int\sf {a}^{x} dx+ \displaystyle\int\sf {(ea)}^{x} dx+ \displaystyle\int\sf sin \: ax \: dx \\ \\ \end{gathered} [/tex]
[tex]\begin{gathered}\sf \: = \: \dfrac{{x}^{a + 1}}{a + 1} + \dfrac{{a}^{x}}{loga} + \dfrac{{(ae)}^{x}}{log(ae)} - \dfrac{cos \: ax}{a} + c \\ \\ \end{gathered}
[/tex]
[tex]\rule{195pt}{2pt}[/tex]
Formula Used :-
[tex]\begin{gathered}\boxed{ \bf{ \:\displaystyle\int\sf {x}^{n} \: dx \: = \: \frac{ {x}^{n + 1} }{n + 1} + c \: }} \\ \\ \end{gathered} [/tex]
[tex]\begin{gathered}\boxed{ \bf{ \:\displaystyle\int\bf {a}^{x} \: dx \: = \: \frac{{a}^{x}}{loga} + c\: }} \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered}\boxed{ \sf{ \:\displaystyle\int\bf {e}^{x} \: dx \: = \: {e}^{x} + c \: }} \\ \\ \end{gathered}
[tex]\begin{gathered}\boxed{ \bf{ \:\displaystyle\int\sf sin(ax + b) \: = \: - \: \dfrac{cos(ax + b)}{a} \: + \: c \: }} \\ \\ \end{gathered} [/tex]
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Answers & Comments
Verified answer
[tex]\large\underline{\sf{Solution-}}[/tex]
[tex]\sf \: \displaystyle\int\sf ( {x}^{a} + {a}^{x} + {e}^{x} {a}^{x} + sin \: ax) \: dx \\ \\ [/tex]
[tex]\sf \: = \: \displaystyle\int\sf {x}^{a}dx + \displaystyle\int\sf {a}^{x} dx+ \displaystyle\int\sf {(ea)}^{x} dx+ \displaystyle\int\sf sin \: ax \: dx \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{{x}^{a + 1}}{a + 1} + \dfrac{{a}^{x}}{loga} + \dfrac{{(ae)}^{x}}{log(ae)} - \dfrac{cos \: ax}{a} + c \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Formulae Used :-
[tex]\boxed{ \sf{ \:\displaystyle\int\sf {x}^{n} \: dx \: = \: \frac{ {x}^{n + 1} }{n + 1} + c \: }} \\ \\ [/tex]
[tex]\boxed{ \sf{ \:\displaystyle\int\sf {a}^{x} \: dx \: = \: \frac{{a}^{x}}{loga} + c\: }} \\ \\ [/tex]
[tex]\boxed{ \sf{ \:\displaystyle\int\sf {e}^{x} \: dx \: = \: {e}^{x} + c \: }} \\ \\ [/tex]
[tex]\boxed{ \sf{ \:\displaystyle\int\sf sin(ax + b) \: = \: - \: \dfrac{cos(ax + b)}{a} \: + \: c \: }} \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
[tex] {{ \mathfrak{Additional\:Information}}}[/tex]
[tex]\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}[/tex]
Step-by-step explanation:
[tex]\large\underline{\bf{Solution}} [/tex]
[tex]\begin{gathered}\sf \: \displaystyle\int\sf ( {x}^{a} + {a}^{x} + {e}^{x} {a}^{x} + sin \: ax) \: dx \\ \\ \end{gathered} [/tex]
[tex]\begin{gathered}\sf \: = \: \displaystyle\int\sf {x}^{a}dx + \displaystyle\int\sf {a}^{x} dx+ \displaystyle\int\sf {(ea)}^{x} dx+ \displaystyle\int\sf sin \: ax \: dx \\ \\ \end{gathered} [/tex]
[tex]\begin{gathered}\sf \: = \: \dfrac{{x}^{a + 1}}{a + 1} + \dfrac{{a}^{x}}{loga} + \dfrac{{(ae)}^{x}}{log(ae)} - \dfrac{cos \: ax}{a} + c \\ \\ \end{gathered}
[/tex]
[tex]\rule{195pt}{2pt}[/tex]
Formula Used :-
[tex]\begin{gathered}\boxed{ \bf{ \:\displaystyle\int\sf {x}^{n} \: dx \: = \: \frac{ {x}^{n + 1} }{n + 1} + c \: }} \\ \\ \end{gathered} [/tex]
[tex]\begin{gathered}\boxed{ \bf{ \:\displaystyle\int\bf {a}^{x} \: dx \: = \: \frac{{a}^{x}}{loga} + c\: }} \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered}\boxed{ \sf{ \:\displaystyle\int\bf {e}^{x} \: dx \: = \: {e}^{x} + c \: }} \\ \\ \end{gathered}
[/tex]
[tex]\begin{gathered}\boxed{ \bf{ \:\displaystyle\int\sf sin(ax + b) \: = \: - \: \dfrac{cos(ax + b)}{a} \: + \: c \: }} \\ \\ \end{gathered} [/tex]