What is the integral from 1 to 4 of sqrt(1+x^2) ?

The integral from 1 to 4 of sqrt(1+x^2) is 2sqrt(17)+(-sqrt(2)-ln(1+sqrt(2))+ln(4+sqrt(17)))/2

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integral from 1 to 4 of sqrt(1+x^2)

\int_{1}^{4} 1 + x^{2} d x

217 + \frac{- 2 - ln ( 1 + 2 ) + ln ( 4 + 17 )}{2}

Decimal

8.14577 \dots

Solution steps

\int_{1}^{4} 1 + x^{2} d x

Apply Trigonometric Substitution

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

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

Apply Integral Reduction

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

\int_{\frac{π}{4}}^{a r c t a n (4)} sec (u) d u = ln (4 + 17) - ln (1 + 2)

= [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{\frac{π}{4}}^{a r c t a n (4)} + \frac{1}{2} (ln (4 + 17) - ln (1 + 2))

Simplify

[\frac{sec ^{2} ( u ) sin ( u )}{2}]_{\frac{π}{4}}^{a r c t a n (4)} + \frac{1}{2} (ln (4 + 17) - ln (1 + 2)) : [\frac{1}{2} sec (u) tan (u)]_{\frac{π}{4}}^{a r c t a n (4)} + \frac{1}{2} (ln (4 + 17) - ln (1 + 2))

= [\frac{1}{2} sec (u) tan (u)]_{\frac{π}{4}}^{a r c t a n (4)} + \frac{1}{2} (ln (4 + 17) - ln (1 + 2))

Compute the boundaries:

217 - \frac{1}{2}

= 217 - \frac{1}{2} + \frac{1}{2} (ln (4 + 17) - ln (1 + 2))

Simplify

= 217 + \frac{- 2 - ln ( 1 + 2 ) + ln ( 4 + 17 )}{2}

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What is the integral from 1 to 4 of sqrt(1+x^2) ?
The integral from 1 to 4 of sqrt(1+x^2) is 2sqrt(17)+(-sqrt(2)-ln(1+sqrt(2))+ln(4+sqrt(17)))/2