What is the integral from 0 to 5 of sqrt(1+t^2) ?

The integral from 0 to 5 of sqrt(1+t^2) is (5sqrt(13))/(sqrt(2))+1/2 ln(5+sqrt(26))

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integral from 0 to 5 of sqrt(1+t^2)

\int_{0}^{5} 1 + t^{2} d t

\frac{5 13}{2} + \frac{1}{2} ln (5 + 26)

Decimal

13.90376 \dots

Solution steps

\int_{0}^{5} 1 + t^{2} d t

Apply Trigonometric Substitution

: \int_{0}^{a r c t a n (5)} sec^{3} (u) d u

= \int_{0}^{a r c t a n (5)} sec^{3} (u) d u

Apply Integral Reduction

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

\int_{0}^{a r c t a n (5)} sec (u) d u = ln (5 + 26)

= [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{a r c t a n (5)} + \frac{1}{2} ln (5 + 26)

Simplify

[\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{a r c t a n (5)} + \frac{1}{2} ln (5 + 26) : [\frac{1}{2} sec (u) tan (u)]_{0}^{a r c t a n (5)} + \frac{1}{2} ln (5 + 26)

= [\frac{1}{2} sec (u) tan (u)]_{0}^{a r c t a n (5)} + \frac{1}{2} ln (5 + 26)

Compute the boundaries:

\frac{5 13}{2}

= \frac{5 13}{2} + \frac{1}{2} ln (5 + 26)

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What is the integral from 0 to 5 of sqrt(1+t^2) ?
The integral from 0 to 5 of sqrt(1+t^2) is (5sqrt(13))/(sqrt(2))+1/2 ln(5+sqrt(26))