Solve ∫ (t+3/t^2+4t+5)^2

$$\int ( \frac { t + 3 } { t ^ { 2 } + 4 t + 5 } ) ^ { 2 }$$

Evaluate

$$\frac{xt^{2}+6tx+9x}{\left(t^{2}+4t+5\right)^{2}}+С$$

Solution Steps

Find the integral of \left(\frac{t+3}{t^{2}+4t+5}\right)^{2} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\left(\frac{t+3}{t^{2}+4t+5}\right)^{2}x$$

Simplify.

$$\frac{\left(t+3\right)^{2}x}{\left(t^{2}+4t+5\right)^{2}}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\frac{\left(t+3\right)^{2}x}{\left(t^{2}+4t+5\right)^{2}}+С$$

Differentiate w.r.t. x

$$\left(\frac{t+3}{t^{2}+4t+5}\right)^{2}$$