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

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

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

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

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

Solution steps

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

Apply Trigonometric Substitution

: \int_{\frac{π}{4}}^{a r c t a n (2)} sec^{3} (u) d u

= \int_{\frac{π}{4}}^{a r c t a n (2)} sec^{3} (u) d u

Apply Integral Reduction

= [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{\frac{π}{4}}^{a r c t a n (2)} + \frac{1}{2} \cdot \int_{\frac{π}{4}}^{a r c t a n (2)} sec (u) d u

\int_{\frac{π}{4}}^{a r c t a n (2)} sec (u) d u = ln (2 + 5) - ln (1 + 2)

= [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{\frac{π}{4}}^{a r c t a n (2)} + \frac{1}{2} (ln (2 + 5) - ln (1 + 2))

Simplify

[\frac{sec ^{2} ( u ) sin ( u )}{2}]_{\frac{π}{4}}^{a r c t a n (2)} + \frac{1}{2} (ln (2 + 5) - ln (1 + 2)) : [\frac{1}{2} sec (u) tan (u)]_{\frac{π}{4}}^{a r c t a n (2)} + \frac{1}{2} (ln (2 + 5) - ln (1 + 2))

= [\frac{1}{2} sec (u) tan (u)]_{\frac{π}{4}}^{a r c t a n (2)} + \frac{1}{2} (ln (2 + 5) - ln (1 + 2))

Compute the boundaries:

5 - \frac{1}{2}

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

Simplify

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

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