What is the integral from-10 to x of sqrt(t^2+11) ?

The integral from-10 to x of sqrt(t^2+11) is

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integral from-10 to x of sqrt(t^2+11)

\int_{- 10}^{x} t^{2} + 11 d t

5111 + \frac{x x ^{2} + 11 + 11 ln \frac{x}{11} + \frac{11 + x ^{2}}{11} - 11 ln ( - \frac{10}{11} + \frac{111}{11} )}{2}

Solution steps

\int_{- 10}^{x} t^{2} + 11 d t

Apply Trigonometric Substitution

: \int_{- a r c t a n (\frac{10}{11})}^{a r c t a n (\frac{x}{11})} 11 sec^{3} (u) d u

= \int_{- a r c t a n (\frac{10}{11})}^{a r c t a n (\frac{x}{11})} 11 sec^{3} (u) d u

Take the constant out:

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

= 11 \cdot \int_{- a r c t a n (\frac{10}{11})}^{a r c t a n (\frac{x}{11})} sec^{3} (u) d u

Apply Integral Reduction

= 11 [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{- a r c t a n (\frac{10}{11})}^{a r c t a n (\frac{x}{11})} + \frac{1}{2} \cdot \int_{- a r c t a n (\frac{10}{11})}^{a r c t a n (\frac{x}{11})} sec (u) d u

\int_{- a r c t a n (\frac{10}{11})}^{a r c t a n (\frac{x}{11})} sec (u) d u = ln \frac{x}{11} + \frac{11 + x ^{2}}{11} - ln (- \frac{10}{11} + \frac{111}{11})

= 11 [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{- a r c t a n (\frac{10}{11})}^{a r c t a n (\frac{x}{11})} + \frac{1}{2} (ln \frac{x}{11} + \frac{11 + x ^{2}}{11} - ln (- \frac{10}{11} + \frac{111}{11}))

Simplify

11 [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{- a r c t a n (\frac{10}{11})}^{a r c t a n (\frac{x}{11})} + \frac{1}{2} (ln \frac{x}{11} + \frac{11 + x ^{2}}{11} - ln (- \frac{10}{11} + \frac{111}{11})) : 11 [\frac{1}{2} sec (u) tan (u)]_{- a r c t a n (\frac{10}{11})}^{a r c t a n (\frac{x}{11})} + \frac{11}{2} ln \frac{x}{11} + \frac{11 + x ^{2}}{11} - \frac{11}{2} ln (- \frac{10}{11} + \frac{111}{11})

= 11 [\frac{1}{2} sec (u) tan (u)]_{- a r c t a n (\frac{10}{11})}^{a r c t a n (\frac{x}{11})} + \frac{11}{2} ln \frac{x}{11} + \frac{11 + x ^{2}}{11} - \frac{11}{2} ln (- \frac{10}{11} + \frac{111}{11})

Compute the boundaries:

\frac{x 11 + x ^{2}}{22} + \frac{5 111}{11}

= 11 (\frac{x 11 + x ^{2}}{22} + \frac{5 111}{11}) + \frac{11}{2} ln \frac{x}{11} + \frac{11 + x ^{2}}{11} - \frac{11}{2} ln (- \frac{10}{11} + \frac{111}{11})

Simplify

= 5111 + \frac{x x ^{2} + 11 + 11 ln \frac{x}{11} + \frac{11 + x ^{2}}{11} - 11 ln ( - \frac{10}{11} + \frac{111}{11} )}{2}

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What is the integral from-10 to x of sqrt(t^2+11) ?
The integral from-10 to x of sqrt(t^2+11) is