What is the integral from 0 to 4 of sqrt(x^2+4) ?

The integral from 0 to 4 of sqrt(x^2+4) is 4(sqrt(5)+1/2 ln(2+sqrt(5)))

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integral from 0 to 4 of sqrt(x^2+4)

\int_{0}^{4} x^{2} + 4 d x

4 (5 + \frac{1}{2} ln (2 + 5))

Decimal

11.83154 \dots

Solution steps

\int_{0}^{4} x^{2} + 4 d x

Apply Trigonometric Substitution

: \int_{0}^{a r c t a n (2)} 4 sec^{3} (u) d u

= \int_{0}^{a r c t a n (2)} 4 sec^{3} (u) d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int_{0}^{a r c t a n (2)} sec^{3} (u) d u

Apply Integral Reduction

= 4 ([\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{a r c t a n (2)} + \frac{1}{2} \cdot \int_{0}^{a r c t a n (2)} sec (u) d u)

\int_{0}^{a r c t a n (2)} sec (u) d u = ln (2 + 5)

= 4 ([\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{a r c t a n (2)} + \frac{1}{2} ln (2 + 5))

Simplify

4 ([\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{a r c t a n (2)} + \frac{1}{2} ln (2 + 5)) : 4 ([\frac{1}{2} sec (u) tan (u)]_{0}^{a r c t a n (2)} + \frac{1}{2} ln (2 + 5))

= 4 ([\frac{1}{2} sec (u) tan (u)]_{0}^{a r c t a n (2)} + \frac{1}{2} ln (2 + 5))

Compute the boundaries:

5

= 4 (5 + \frac{1}{2} ln (2 + 5))

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What is the integral from 0 to 4 of sqrt(x^2+4) ?
The integral from 0 to 4 of sqrt(x^2+4) is 4(sqrt(5)+1/2 ln(2+sqrt(5)))