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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

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

Decimal

2.39052 \dots

Solution steps

\int_{0}^{3} 1 + x^{2} 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(1+x^2) ?
The integral from 0 to sqrt(3 of)sqrt(1+x^2) is sqrt(3)+1/2 ln(2+sqrt(3))