What is the integral from 0 to 2 of sqrt(4x^2-8x+5) ?

The integral from 0 to 2 of sqrt(4x^2-8x+5) is sqrt(5)+1/4 ln(2+sqrt(5))-1/4 ln(-2+sqrt(5))

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integral from 0 to 2 of sqrt(4x^2-8x+5)

\int_{0}^{2} 4 x^{2} - 8 x + 5 d x

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

Decimal Notation

2.95788 \dots

Solution steps

\int_{0}^{2} 4 x^{2} - 8 x + 5 d x

Complete the square

4 x^{2} - 8 x + 5 : 4 (x - 1)^{2} + 1

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

Apply u-substitution

: \int_{- 1}^{1} 4 u^{2} + 1 d u

= \int_{- 1}^{1} 4 u^{2} + 1 d u

Apply Trigonometric Substitution

: \int_{- a r c t a n (2)}^{a r c t a n (2)} \frac{1}{2} sec^{3} (v) d v

= \int_{- a r c t a n (2)}^{a r c t a n (2)} \frac{1}{2} sec^{3} (v) d v

Take the constant out:

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

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

Apply Integral Reduction

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

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

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

Simplify

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

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

Compute the boundaries:

25

= \frac{1}{2} (25 + \frac{1}{2} (ln (2 + 5) - ln (- 2 + 5)))

Simplify

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

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What is the integral from 0 to 2 of sqrt(4x^2-8x+5) ?
The integral from 0 to 2 of sqrt(4x^2-8x+5) is sqrt(5)+1/4 ln(2+sqrt(5))-1/4 ln(-2+sqrt(5))