Solve ∫ (from 0 to 3) of int_0^x(3t^2+t-1)dt wrt x

$$\int _ { 0 } ^ { 3 } \int _ { 0 } ^ { x } ( 3 t ^ { 2 } + t - 1 ) d t d x$$

Evaluate

$$\frac{81}{4}=20.25$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \int _{0}^{x}3t^{2}+t-1\mathrm{d}t\mathrm{d}x$$

Simplify.

$$\frac{x^{4}}{4}+\frac{x^{3}}{6}-\frac{x^{2}}{2}$$

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.

$$\frac{3^{4}}{4}+\frac{3^{3}}{6}-\frac{3^{2}}{2}-\left(\frac{0^{4}}{4}+\frac{0^{3}}{6}-\frac{0^{2}}{2}\right)$$

Simplify.

$$\frac{81}{4}$$