What is the integral from 0 to 2 of sqrt(x^2+1) ?

The integral from 0 to 2 of sqrt(x^2+1) is sqrt(5)+1/2 ln(2+sqrt(5))

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integral from 0 to 2 of sqrt(x^2+1)

\int_{0}^{2} x^{2} + 1 d x

5 + \frac{1}{2} ln (2 + 5)

Decimal

2.95788 \dots

Solution steps

\int_{0}^{2} x^{2} + 1 d x

Apply Trigonometric Substitution

= \int_{0}^{a r c t a n (2)} sec^{3} (u) d u

Apply Integral Reduction

= [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{a r c t a n (2)} + \frac{1}{2} \cdot \int_{0}^{a r c t a n (2)} sec (u) d u

\int_{0}^{a r c t a n (2)} sec (u) d u = ln (2 + 5)

= [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{a r c t a n (2)} + \frac{1}{2} ln (2 + 5)

Simplify

[\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{a r c t a n (2)} + \frac{1}{2} ln (2 + 5) : [\frac{1}{2} sec (u) tan (u)]_{0}^{a r c t a n (2)} + \frac{1}{2} ln (2 + 5)

= [\frac{1}{2} sec (u) tan (u)]_{0}^{a r c t a n (2)} + \frac{1}{2} ln (2 + 5)

Compute the boundaries:

5

= 5 + \frac{1}{2} ln (2 + 5)

What is the integral from 0 to 2 of sqrt(x^2+1) ?
The integral from 0 to 2 of sqrt(x^2+1) is sqrt(5)+1/2 ln(2+sqrt(5))