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

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

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

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

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

Decimal

1.14779 \dots

Solution steps

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

Apply Trigonometric Substitution

= \int_{0}^{\frac{π}{4}} sec^{3} (u) d u

Apply Integral Reduction

= [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{\frac{π}{4}} + \frac{1}{2} \cdot \int_{0}^{\frac{π}{4}} sec (u) d u

\int_{0}^{\frac{π}{4}} sec (u) d u = ln (1 + 2)

= [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{\frac{π}{4}} + \frac{1}{2} ln (1 + 2)

Simplify

[\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{\frac{π}{4}} + \frac{1}{2} ln (1 + 2) : [\frac{1}{2} sec (u) tan (u)]_{0}^{\frac{π}{4}} + \frac{1}{2} ln (1 + 2)

= [\frac{1}{2} sec (u) tan (u)]_{0}^{\frac{π}{4}} + \frac{1}{2} ln (1 + 2)

Compute the boundaries:

\frac{1}{2}

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

Simplify

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

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