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$\int \frac{{mx}^{m + 2 n - 1} - {nx}^{n - 1}}{x^{2 m + 2 n} + 2 x^{m + n} + 1} dx$ is equal to

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xmxm+n+1+C

xnxm+n+1+C

xm+n−1xm+n+1+C

-xnxm+n+1+C

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Correct option is D

Let I=∫mxm+2n−1−nxn−1x2m+2n+2xm+n+1dx=∫mxm+2n−1−nxn−11+xm+n2dx∴I=∫mxm−1−nxn+1xm+1xn2dxLet xm+1xn=t⇒mxm−1−nxn+1dx=dt∴I=∫1t2dt=−1t+C=−1xm+1xn+C=−xnxm+n+1+C

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∫mxm+2n−1−nxn−1x2m+2n+2xm+n+1dx is equal to

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$\int \frac{{mx}^{m + 2 n - 1} - {nx}^{n - 1}}{x^{2 m + 2 n} + 2 x^{m + n} + 1} dx$ is equal to