Question

Solve following integral? `∫sin^-1(2x)dx`

Answer 1

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`"Using "color(blue)"Integration by Parts :"`
`color(blue)(int(u*v)dx=uintvdx-int(u'intvdx)dx`

Explanation:

`color(red)((a)I=intsin^-1(2x)dx`
,
Let, `sin^-1 2x=t=>2x=sint=>x=1/2sint=>dx=1/2costdt`

`=>I=1/2int t*costdt...tou=t and v=cost`

`"Using "color(blue)"Integration by Parts :"`

`I=1/2[tsint-intsintdt]=1/2[tsint+cost]+c`

`I=1/2[tsint+sqrt(1-sin^2t)]+c`

Subst.back `t=sin^-1(2x)and sint=2x`

`color(red)(I=1/2[2x*sin^-1(2x)+sqrt(1-4x^2)]+c`

`color(violet)((b)I=intxln(3x)dx...tou=ln3x and v=x`

`"Using "color(blue)"Integration by Parts :"`

`I=ln(3x)*x^2/2-int1/(3x)xx3xxx^2/2dx`

`I=x^2/2ln(3x)-1/2intxdx=x^2/2ln(3x)-1/2(x^2/2)+c`

`=>color(violet)(I=x^2/2[ln3x-1/2]+c`

`color(orange)((c)intx^2 e^xdx...tou=x^2 and v=e^x`

`"Using "color(blue)"Integration by Parts :"`

`I=x^2e^x-int2xe^xdx=x^2e^x-2intxe^x`

`"Using "color(blue)"Integration by Parts :"`

`I=x^2e^x-2[xe^x-inte^xdx]`

`color(orange)(I=x^2e^x-2xe^x+2e^x+c`