What is the integral of sqrt(x^2+2x+5) ?

The integral of sqrt(x^2+2x+5) is 4(1/8 (x+1)sqrt(x^2+2x+5)+1/2 ln((|x+1+sqrt(x^2+2x+5)|)/2))+C

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integral of sqrt(x^2+2x+5)

\int x^{2} + 2 x + 5 d x

4 (\frac{1}{8} (x + 1) x^{2} + 2 x + 5 + \frac{1}{2} ln (\frac{x + 1 + x ^{2} + 2 x + 5}{2})) + C

Solution steps

\int x^{2} + 2 x + 5 d x

Complete the square

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

= \int (x + 1)^{2} + 4 d x

Apply u-substitution

: \int u^{2} + 4 d u

= \int u^{2} + 4 d u

Apply Trigonometric Substitution

: \int 4 sec^{3} (v) d v

= \int 4 sec^{3} (v) d v

Take the constant out:

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

= 4 \cdot \int sec^{3} (v) d v

Apply Integral Reduction

= 4 (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} \cdot \int sec (v) d v)

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= 4 (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

Substitute back

= 4 (\frac{sec ^{2} ( arctan ( \frac{1}{2} ( x + 1 ) ) ) sin ( arctan ( \frac{1}{2} ( x + 1 ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{2} (x + 1))) + sec (arctan (\frac{1}{2} (x + 1))))

Simplify

4 (\frac{sec ^{2} ( arctan ( \frac{1}{2} ( x + 1 ) ) ) sin ( arctan ( \frac{1}{2} ( x + 1 ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{2} (x + 1))) + sec (arctan (\frac{1}{2} (x + 1)))) : 4 (\frac{1}{8} (x + 1) x^{2} + 2 x + 5 + \frac{1}{2} ln (\frac{x + 1 + x ^{2} + 2 x + 5}{2}))

= 4 (\frac{1}{8} (x + 1) x^{2} + 2 x + 5 + \frac{1}{2} ln (\frac{x + 1 + x ^{2} + 2 x + 5}{2}))

Add a constant to the solution

= 4 (\frac{1}{8} (x + 1) x^{2} + 2 x + 5 + \frac{1}{2} ln (\frac{x + 1 + x ^{2} + 2 x + 5}{2})) + C

f (x) = x^{2} + 8, (4, 24)

\frac{d}{d x} (\frac{3 - 2 x}{3 + 2 x})

\int x^{4} sin (3 x) d x

\int - (x^{2} + 2) e^{x} d x

\frac{\partial}{\partial x} (3^{2 y})

What is the integral of sqrt(x^2+2x+5) ?
The integral of sqrt(x^2+2x+5) is 4(1/8 (x+1)sqrt(x^2+2x+5)+1/2 ln((|x+1+sqrt(x^2+2x+5)|)/2))+C