Solve ∫ t^2gdg

$$\int t ^ { 2 } g d g$$

Evaluate

$$\frac{\left(gt\right)^{2}}{2}+С$$

Solution Steps

Factor out the constant using \int af\left(g\right)\mathrm{d}g=a\int f\left(g\right)\mathrm{d}g.

$$t^{2}\int g\mathrm{d}g$$

Since \int g^{k}\mathrm{d}g=\frac{g^{k+1}}{k+1} for k\neq -1, replace \int g\mathrm{d}g with \frac{g^{2}}{2}.

$$t^{2}\times \frac{g^{2}}{2}$$

Simplify.

$$\frac{t^{2}g^{2}}{2}$$

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

$$\frac{t^{2}g^{2}}{2}+С$$

Differentiate w.r.t. t

$$tg^{2}$$