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

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lemjay eNotes educator| Certified Educator

`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)`


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`

mathsworkmusic eNotes educator| Certified Educator

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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