What is the integral from 0 to sqrt(3 of)sqrt(x^2+1) ?

The integral from 0 to sqrt(3 of)sqrt(x^2+1) is sqrt(3)+1/2 ln(2+sqrt(3))

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integral from 0 to sqrt(3 of)sqrt(x^2+1)

\int_{0}^{3} x^{2} + 1 d x

3 + \frac{1}{2} ln (2 + 3)

Decimal

2.39052 \dots

Solution steps

\int_{0}^{3} x^{2} + 1 d x

Apply Trigonometric Substitution

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

Apply Integral Reduction

= [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{\frac{π}{3}} + \frac{1}{2} \cdot \int_{0}^{\frac{π}{3}} sec (u) d u

\int_{0}^{\frac{π}{3}} sec (u) d u = ln (2 + 3)

= [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{\frac{π}{3}} + \frac{1}{2} ln (2 + 3)

Simplify

[\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{\frac{π}{3}} + \frac{1}{2} ln (2 + 3) : [\frac{1}{2} sec (u) tan (u)]_{0}^{\frac{π}{3}} + \frac{1}{2} ln (2 + 3)

= [\frac{1}{2} sec (u) tan (u)]_{0}^{\frac{π}{3}} + \frac{1}{2} ln (2 + 3)

Compute the boundaries:

3

= 3 + \frac{1}{2} ln (2 + 3)

What is the integral from 0 to sqrt(3 of)sqrt(x^2+1) ?
The integral from 0 to sqrt(3 of)sqrt(x^2+1) is sqrt(3)+1/2 ln(2+sqrt(3))