Question

Integral..Could you help me,please?

Answer 1

Answer:

as we know, the integral of $x^{n}$ is $\frac{x^{n+1} }{n+1}$

so, the integral of x.$2^{x}$ will be found as follows:

here, we will use a trick called 'integration by parts'

let x = u and $2^{x}$ = v

∫uv dx = u∫v dx - ∫[(du/dx)* ∫v dx] dx

∫x.$2^{x}$ dx = x∫$2^{x}$ - ∫[(dx/dx) * ∫$2^{x}$ dx ] dx

∫x.$2^{x}$ dx = x*$\frac{2^{x+ 1}}{x+ 1}$ - 1 * $\frac{2^{x + 1} }{x + 1}$

= $\frac{2^{x} }{x + 1}$( x - 1 )