Integration by Parts in Zee's QFT: Understanding Eq. (14) to Eq. (15)

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Homework Statement

I'm studying from Zee's QFT in a nutshell. On page 21, I don't understand how he uses integration by parts to get from Eq (14) to Eq (15), ie from

Z = \int D \varphi e^{i \int d^4 x \{ \frac{1}{2}[(\partial \varphi)^2 - m^2 \varphi^2] + J\varphi \}}

to

Z = \int D \varphi e^{i \int d^4 x [-\frac{1}{2}(\partial^2+m^2)\varphi + J\varphi]}.

Is he suggesting that \int d^4x \varphi^2 = \int d^4x \varphi and \int d^4x (\partial\varphi)^2 = \int d^4x( -\partial^2 \varphi)? If so, I'm failing to see why this should be the case.

 

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Sigh. Nevermind, there was a typo in my second integral, Eq. (15) is actually

Z = \int D \varphi e^{i \int d^4 x [-\frac{1}{2}\varphi(\partial^2+m^2)\varphi + J\varphi]} which can be obtained easily by integration by parts on the (\partial \varphi)^2 term:

\int d^4 x\, (\partial \varphi)^2 = -\int d^4x\, \varphi \partial^2\varphi.