Solve ∫ (from 0 to 5) of sqrt{1+y^2 wrt x

$$\int_{ 0 }^{ 5 } \sqrt{ 1+ { y }^{ 2 } } d x$$

Evaluate

$$5\sqrt{y^{2}+1}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \sqrt{1+y^{2}}\mathrm{d}x$$

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

$$\sqrt{1+y^{2}}x$$

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.

$$\left(1+y^{2}\right)^{\frac{1}{2}}\times 5-\left(1+y^{2}\right)^{\frac{1}{2}}\times 0$$

Simplify.

$$5\sqrt{1+y^{2}}$$

Differentiate w.r.t. y

$$\frac{5y}{\sqrt{y^{2}+1}}$$