Solve ∫ (from 0 to 1) of sqrt{1+(2t)^2}

$$\int _ { 0 } ^ { 1 } \sqrt { 1 + ( 2 t ) ^ { 2 } }$$

Evaluate

$$\sqrt{4t^{2}+1}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \sqrt{1+\left(2t\right)^{2}}\mathrm{d}x$$

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

$$\sqrt{1+\left(2t\right)^{2}}x$$

Simplify.

$$\sqrt{1+4t^{2}}x$$

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.

$$\left(1+4t^{2}\right)^{\frac{1}{2}}\times 1-\left(1+4t^{2}\right)^{\frac{1}{2}}\times 0$$

Simplify.

$$\sqrt{1+4t^{2}}$$

Differentiate w.r.t. t

$$\frac{4t}{\sqrt{4t^{2}+1}}$$