A matrix exponential identity

Question

In a physics textbook, the following identity was claimed to hold $$ e^{A+B} = e^{A} + \int_0^1 e^{\lambda(A+B)} B e^{(1-\lambda)A} d\lambda $$ with $A$ and $B$ both being complex matrices which do not necessarily commute. I have tried to convince myself of this identity by considering $$ \begin{align} \frac{d}{d\lambda} e^{A+\lambda B} &= B e^{A+\lambda B} \left( = e^{A+\lambda B}B \right)\\ e^{A+B} - e^A &= \int_0^1 B e^{A+\lambda B} d\lambda \\ e^{A+B} &= e^A + \int_0^1 B e^{\lambda (A+B) + (1-\lambda) A} d\lambda \\ &\stackrel{?}{=} e^A + \int_0^1 B e^{\lambda (A+B)} e^{(1-\lambda) A} d\lambda \\ &= e^A + \int_0^1 e^{\lambda (A+B)} B e^{(1-\lambda) A} d\lambda \\ \end{align} $$ where I have marked the dubious step with "?". Am I wrong to think this step incorrect, or is there an alternative way to the result?