Solve ∫ (from a to x) of S^2e^lnsds

$$\int _ { a } ^ { x } S ^ { 2 } e ^ { \ln s } d s$$

Evaluate

$$\frac{\left(Sx\right)^{2}-\left(Sa\right)^{2}}{2}$$

Solution Steps

Apply rule a^{\log_{a}\left(b\right)}=b, where a=e and b=s.

$$\int _{a}^{x}S^{2}s\mathrm{d}s$$

Evaluate the indefinite integral first.

$$\int S^{2}s\mathrm{d}s$$

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

$$S^{2}\int s\mathrm{d}s$$

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

$$S^{2}\times \frac{s^{2}}{2}$$

Simplify.

$$\frac{S^{2}s^{2}}{2}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{1}{2}S^{2}x^{2}-\frac{1}{2}S^{2}a^{2}$$

Simplify.

$$\frac{\left(x-a\right)\left(x+a\right)S^{2}}{2}$$

Differentiate w.r.t. x

$$xS^{2}$$