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

The integral from 0 to 2 of sqrt(1+(-2x)^2) is sqrt(17)+1/4 ln(4+sqrt(17))

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

\int_{0}^{2} 1 + (- 2 x)^{2} d x

17 + \frac{1}{4} ln (4 + 17)

Decimal Notation

4.64678 \dots

Solution steps

\int_{0}^{2} 1 + (- 2 x)^{2} d x

Simplify

1 + (- 2 x)^{2} : 1 + 4 x^{2}

= \int_{0}^{2} 1 + 4 x^{2} d x

Apply Trigonometric Substitution

: \int_{0}^{a r c t a n (4)} \frac{1}{2} sec^{3} (u) d u

= \int_{0}^{a r c t a n (4)} \frac{1}{2} sec^{3} (u) d u

Take the constant out:

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

= \frac{1}{2} \cdot \int_{0}^{a r c t a n (4)} sec^{3} (u) d u

Apply Integral Reduction

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

\int_{0}^{a r c t a n (4)} sec (u) d u = ln (4 + 17)

= \frac{1}{2} ([\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{a r c t a n (4)} + \frac{1}{2} ln (4 + 17))

Simplify

\frac{1}{2} ([\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{a r c t a n (4)} + \frac{1}{2} ln (4 + 17)) : \frac{1}{2} ([\frac{1}{2} sec (u) tan (u)]_{0}^{a r c t a n (4)} + \frac{1}{2} ln (4 + 17))

= \frac{1}{2} ([\frac{1}{2} sec (u) tan (u)]_{0}^{a r c t a n (4)} + \frac{1}{2} ln (4 + 17))

Compute the boundaries:

217

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

Simplify

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

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What is the integral from 0 to 2 of sqrt(1+(-2x)^2) ?
The integral from 0 to 2 of sqrt(1+(-2x)^2) is sqrt(17)+1/4 ln(4+sqrt(17))