is $\int_{0}^{t} dB_s = B_t$?

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$(B_t)_{t \geq 0}$ is a standard Brownian motion, first of all I know that they are equal in distribution but as I was solving the Geometric Brownian motion SDE I got a solution of the form :

$$X_t = X_0 \exp((r - \frac{\sigma^2}{2})t + \sigma\int_{0}^{t} dB_s)$$

but the solution that is written in the lecture notes I'm reading from is of the form :

$$X_t = X_0 \exp((r - \frac{\sigma^2}{2})t + \sigma B_t)$$

Am I wrong and should review my work or are the two forms the same thing ?