Question

Integrate (x/2+5secxtanx-6csc^2x+4^x+8)?

Answer 1

`I=x^2/4+5secx+6cotx+4^x/lnx+8x+C`

Explanation:

Formulae:
`color(red)((1)int(x^n)dx=(x^(n+1))/(n+1)+c)`
`color(red)((2)int(secthetatantheta)d theta=sectheta+c`
`color(red)((3)int(csc^2theta)d theta=-cottheta+c`
`color(red)((4)int(a^X)dX=a^x/lna+c`
`color(red)((5)intAdx=x+c,where,A` is constant.

`I=int(x/2+5secxtanx-6csc^2x+4^x+8)dx,....to[I]`

#color(red) ((1)=>)int1/2xdx=1/2(x^(1+1))/(1+1)+c_1=1/2*x^2/2+c_1=x^2/4+c_1#
`color(red)((2)=>)int5secxtanx=5secx+c_2`
`color(red)((3)=>)int6csc^2xdx=-6cotx+c_3`
`color(red)((4)=>)int4^xdx=4^x/ln4+c_4`
`color(red)((5)=>)int8dx=8x+c_5`

Substituting all values to above, `[I]` we get

`I=x^2/4+5secx+6cotx+4^x/lnx+8x+C`

where, `C=c_1+c_2+c_3+c_4+c_5`