Solve ∫ (from t to a) of a^2+2x-3

$$\int _ { t } ^ { a } a ^ { 2 } + 2 x - 3$$

Evaluate

$$\left(a-t\right)\left(t+a^{2}+a-3\right)$$

Solution Steps

Evaluate the indefinite integral first.

$$\int a^{2}+2x-3\mathrm{d}x$$

Integrate the sum term by term.

$$\int a^{2}\mathrm{d}x+\int 2x\mathrm{d}x+\int -3\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int a^{2}\mathrm{d}x+2\int x\mathrm{d}x+\int -3\mathrm{d}x$$

Find the integral of a^{2} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$a^{2}x+2\int x\mathrm{d}x+\int -3\mathrm{d}x$$

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

$$a^{2}x+x^{2}+\int -3\mathrm{d}x$$

Find the integral of -3 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$a^{2}x+x^{2}-3x$$

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.

$$a^{2}a+a^{2}-3a-\left(a^{2}t+t^{2}-3t\right)$$

Simplify.

$$\left(-3+a+a^{2}+t\right)\left(a-t\right)$$

Differentiate w.r.t. a

$$-2at+3a^{2}+2a-3$$