Solve ∫ (from 0 to 5) of arctan(t^2+t-1)

$$\int _ { 0 } ^ { 5 } \arctan ( t ^ { 2 } + t - 1 )$$

Evaluate

$$5\arctan(t^{2}+t-1)$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \arctan(t^{2}+t-1)\mathrm{d}x$$

Find the integral of \arctan(t^{2}+t-1) using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\arctan(t^{2}+t-1)x$$

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.

$$5\arctan(t^{2}+t-1)+0\arctan(t^{2}+t-1)$$

Simplify.

$$5\arctan(t^{2}+t-1)$$

Differentiate w.r.t. t

$$\frac{5\left(2t+1\right)}{t^{4}+2t^{3}-t^{2}-2t+2}$$