∫ 2 mx ⋅ 3 nx dx when m , n ∈ N is equal to

Question

∫ 2 mx ⋅ 3 nx dx when m , n ∈ N is equal to (A) (2 mx +3 nx / m λ n 2+ n λ n 3)+ c (B) (e(m / n 2+n λ n 3) x/m λ n 2+n λ n 3)+c (C) (2 mx ⋅ 3 nx /λ n (2 m ⋅ 3 n ))+ c (D) (( mn ) ⋅ 2 x ⋅ 3 x / m λ n 2+ n λ n 3)+ c

Tardigrade Admin · Accepted Answer

2 mx = e mx ln 2 and 3 nx = e nx ln 3.

Q. $\int 2^{m x} \cdot 3^{n x} d x$ when $m, n \in N$ is equal to

$\frac{2 ^{m x} + 3 ^{n x}}{mλn 2 + nλn 3} + c$

$\frac{e ^{(m / n 2 + nλn 3) x}}{mλn 2 + nλn 3} + c$

$\frac{2 ^{m x} \cdot 3 ^{n x}}{λn ( 2 ^{m} \cdot 3 ^{n} )} + c$

$\frac{( mn ) \cdot 2 ^{x} \cdot 3 ^{x}}{mλn 2 + nλn 3} + c$

$2^{m x} = e^{m x l n 2}$ and $3^{n x} = e^{n x l n 3}$ .

Q. ∫2mx⋅3nxdx when m,n∈N is equal to

mλn2+nλn32mx+3nx​+c

mλn2+nλn3e(m/n2+nλn3)x​+c

λn(2m⋅3n)2mx⋅3nx​+c

mλn2+nλn3(mn)⋅2x⋅3x​+c