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Evaluate the integral using substitution

by RoRi

Use the method of substitution to evaluate the following integral:

    \[ \int \frac{(x+1)\, dx}{(x^2+2x+3)^3}. \]

Let

    \[ u = x^2 + 2x + 3 \quad \implies \quad du = 2(x + 1) \, dx. \]

Then, we can evaluate the integral,

    \begin{align*} \int \frac{(x+1) \, dx}{(x^2+ 2x + 3)^3} &= \frac{1}{2} \int u^{-3} \, du \\ &= -\frac{1}{4} u^{-2} + C \\ &= -\frac{1}{4} (x^2 + 2x + 3)^{-2} + C. \end{align*}

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