Question

Evaluate the following integrals.
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Plss answer fast​

Answer 1

$\large\underline{\bold{Given \:Question - }}$

Evaluate the following integral

$\displaystyle\int\sf \bigg[{e}^{3alogx} + {e}^{3xloga}\bigg] \: dx$

$\large\underline{\sf{Solution-}}$

The given integral is

$\rm :\longmapsto\:\displaystyle\int\sf \bigg[{e}^{3alogx} + {e}^{3xloga}\bigg] \: dx$

We know,

$\boxed{ \bf{ \:bloga = log {a}^{b}}}$

So, using this, we get

$\rm \:  =  \: \displaystyle\int\sf \bigg[{e}^{log {x}^{3a} } + {e}^{log {a}^{3x} }\bigg] \: dx$

We know,

$\boxed{ \bf{ \: {e}^{logx} = x \: }}$

$\rm \:  =  \: \displaystyle\int\sf \bigg[ {x}^{3a} + {a}^{3x}\bigg] \: dx$

We know,

$\boxed{ \bf{ \:\displaystyle\int\sf {x}^{n} \: dx \: = \: \frac{ {x}^{n + 1} }{n + 1} \: + \: c \: \: }}$

and

$\boxed{ \bf{ \:\displaystyle\int\sf {a}^{x} \: dx \: = \: \frac{ {a}^{x} }{loga} \: + \: c \: \: }}$

So, using this, we get

$\rm \:  =  \: \dfrac{ {x}^{3a + 1} }{3(3a + 1)} + \dfrac{ {a}^{3x} }{3 \: loga} + c$

$\red{\boxed{ \bf{ \:\displaystyle\int\sf \bigg[{e}^{3alogx} + {e}^{3xloga}\bigg]dx=  \: \dfrac{ {x}^{3a + 1} }{3(3a + 1)} + \dfrac{ {a}^{3x} }{3 \: loga} + c}}}$

Additional Information :-

$\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}$

Answer 2

$</p><p>\large\underline{\bold{Given \:Question - }} </p><p>$

Evaluate the following integral

$\displaystyle\int\sf \bigg[{e}^{3alogx} + {e}^{3xloga}\bigg] \: dx∫[e </p><p>3alogx</p><p> +e </p><p>3xloga</p><p> ]dx$

$\large\underline{\sf{Solution-}} </p><p>$

The given integral is

$</p><p>\rm :\longmapsto\:\displaystyle\int\sf \bigg[{e}^{3alogx} + {e}^{3xloga}\bigg] \: dx:⟼∫[e </p><p>3alogx</p><p> +e </p><p>3xloga</p><p> ]dx</p><p></p><p>$

We know,

$</p><p>\boxed{ \bf{ \:bloga = log {a}^{b}}} </p><p>$

So, using this, we get

$\rm \: = \: \displaystyle\int\sf \bigg[{e}^{log {x}^{3a} } + {e}^{log {a}^{3x} }\bigg] \: dx = ∫[e </p><p>logx </p><p>3a</p><p> </p><p> +e </p><p>loga </p><p>3x</p><p> </p><p> ]dx</p><p>$

We know,

$\boxed{ \bf{ \: {e}^{logx} = x \: }} </p><p>e$

$\rm \: = \: \displaystyle\int\sf \bigg[ {x}^{3a} + {a}^{3x}\bigg] \: dx = ∫[x </p><p>3a</p><p> +a </p><p>3x</p><p> ]dx$

We know,

$\boxed{ \bf{ \:\displaystyle\int\sf {x}^{n} \: dx \: = \: \frac{ {x}^{n + 1} }{n + 1} \: + \: c \: \: }} </p><p></p><p></p><p> </p><p></p><p>and$

$</p><p>\boxed{ \bf{ \:\displaystyle\int\sf {a}^{x} \: dx \: = \: \frac{ {a}^{x} }{loga} \: + \: c \: \: }} </p><p>$

So, using this, we get

$</p><p>\rm \: = \: \dfrac{ {x}^{3a + 1} }{3(3a + 1)} + \dfrac{ {a}^{3x} }{3 \: loga} + c = </p><p>3(3a+1)</p><p>x </p><p>3a+1</p><p> </p><p></p><p> + </p><p>3loga</p><p>a </p><p>3x</p><p> </p><p></p><p> +c$

$\red{\boxed{ \bf{ \:\displaystyle\int\sf \bigg[{e}^{3alogx} + {e}^{3xloga}\bigg]dx= \: \dfrac{ {x}^{3a + 1} }{3(3a + 1)} + \dfrac{ {a}^{3x} }{3 \: loga} + c}}} </p><p>∫[e </p><p>3alogx</p><p> +e </p><p>3xloga</p><p> ]dx= </p><p>3(3a+1)</p><p>x </p><p>3a+1</p><p> </p><p></p><p> + </p><p>3loga</p><p>a </p><p>3x</p><p> </p><p></p><p> +c</p><p></p><p> </p><p></p><p>Additional Information :-</p><p>\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}\end{gathered} </p><p>f(x)</p><p></p><p> </p><p>k</p><p>sinx</p><p>cosx</p><p>sec </p><p>2</p><p> x</p><p>cosec </p><p>2</p><p> x</p><p>secxtanx</p><p>cosecxcotx</p><p>tanx</p><p>x</p><p>1</p><p></p><p> </p><p>e </p><p>x</p><p> </p><p></p><p> </p><p>∫f(x)dx</p><p></p><p> </p><p>kx+c</p><p>−cosx+c</p><p>sinx+c</p><p>tanx+c</p><p>−cotx+c</p><p>secx+c</p><p>−cosecx+c</p><p>logsecx+c</p><p>logx+c</p><p>e </p><p>x</p><p> +c</p><p></p><p> </p><p></p><p>$