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

The integral of 2sqrt(x^2+1) is xsqrt(1+x^2)+ln|x+sqrt(1+x^2)|+C

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integral of 2sqrt(x^2+1)

\int 2 x^{2} + 1 d x

x 1 + x^{2} + ln x + 1 + x^{2} + C

Solution steps

\int 2 x^{2} + 1 d x

Take the constant out:

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

= 2 \cdot \int x^{2} + 1 d x

Apply Trigonometric Substitution

: \int sec^{3} (u) d u

= 2 \cdot \int sec^{3} (u) d u

Apply Integral Reduction

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

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

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

Substitute back

u = arctan (x)

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

Simplify

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

= x 1 + x^{2} + ln x + 1 + x^{2}

Add a constant to the solution

= x 1 + x^{2} + ln x + 1 + x^{2} + C

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What is the integral of 2sqrt(x^2+1) ?
The integral of 2sqrt(x^2+1) is xsqrt(1+x^2)+ln|x+sqrt(1+x^2)|+C