151. $\int \sqrt{x+1} d x$.

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151. $\int \sqrt{x+1} d x$.
Solution. $\int \sqrt{x+1} d x=\left|\begin{array}{c}z=x+1 \\ d z=d x\end{array}\right|=\int \sqrt{z} d z=\int z^{1 / 2} d z=\frac{2}{3} z^{3 / 2}+$ $+C=\frac{2}{3}(x+1) \sqrt{x+1}+C$.

Verification: $d\left(\frac{2}{3}(x+1) \sqrt{x+1}+C\right)=d\left(\frac{2}{3}(x+1)^{3 / 2}+C\right)=\frac{2}{3} \times$ $\times \frac{3}{2}(x+1)^{1 / 2} d x=\sqrt{x+1} d x$.