# Evaluate the indefinite integral int 7^xcos(x)dx

Using integration by parts and that 7^x = e^(ln7^x) = e^(xln7) and d/dx 7^x = ln7e^(xln7) = ln7 (7^x)

  int 7^x cosx dx = 7^x sinx - int ln7 7^x sinx dx

Using integration by parts again

int 7^x sinx dx = -7^x cosx - int (-ln7 7^x...

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Using integration by parts and that 7^x = e^(ln7^x) = e^(xln7) and d/dx 7^x = ln7e^(xln7) = ln7 (7^x)

 int 7^x cosx dx = 7^x sinx - int ln7 7^x sinx dx

Using integration by parts again

int 7^x sinx dx = -7^x cosx - int (-ln7 7^x cosx) dx

implies int 7^x cosx dx = 7^x sinx + 7^x cosx - int ln7 7^x cosx dx

 Gathering terms we have that

(1 + ln7) int 7^x cosx dx = 7^x (sinx + cosx) 

implies int 7^x cosx dx = 7^x ((sinx + cosx))/((1 + ln7))  answer to question

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int 7^x cos(x) dx

To evaluate, apply integration by parts. The formula is int udv=uv-intvdu .

So let,

u=7^x                         and                dv=cos(x) dx

du=7^x ln 7*dx                                   v=int cos(x)dx=sin(x)

Substitute u,v and du to the formula above.

int 7^x cos(x)dx=7^x sin(x) - int sin(x) 7^xln7* dx

= 7^xsin(x)- ln 7 int 7^x sin(x) dx

Use integration by parts again to evaluate the integral part at the right side.

So let,

u=7^x                         and                dv=sin(x)dx

du=7^x ln7* dx                                   v=int sin(x)dx = -cos(x)

Substituting u, v and du to the formula of integration by parts yields,

int 7^xcos(x)dx=7^x sin(x)-ln7[7^x(-cos(x)) - int -cosx*7^xln7*dx]

=7^x sin(x)-ln7[-7^xcos(x)+ln7int7^xcos(x)dx]

=7^x sin(x)+7^x(ln7)cos(x)-(ln 7)^2int 7^xcos(x)dx

Since same integrand appears on both sides of the equation, to combine like terms, move (ln 7)^2 int 7^xcosxdx to the left side of the equation.

int 7^xcos(x) dx + (ln 7)^2int 7^xcos(x)dx = 7^x sin(x) + 7^x(ln7)cos(x)

Then, factor the integral.

(1+(ln 7)^2)int 7^x cos(x) dx =7^x sin(x) + 7^x(ln7)cos(x)

And isolate the integral.

int 7^x cos(x) dx =(7^x sin(x) + 7^x(ln7)cos(x))/(1+ (ln 7)^2)

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Since the given is an indefinite integral, then

int 7^x cos(x) dx = (7^x sin(x) + 7^x(ln7)cos(x))/(1+(ln7)^2) + C`

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