What is the integral of sqrt(4+16x^2) ?

The integral of sqrt(4+16x^2) is xsqrt(4x^2+1)+1/2 ln|2x+sqrt(1+4x^2)|+C

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integral of sqrt(4+16x^2)

\int 4 + 16 x^{2} d x

x 4 x^{2} + 1 + \frac{1}{2} ln 2 x + 1 + 4 x^{2} + C

Solution steps

\int 4 + 16 x^{2} d x

Apply Trigonometric Substitution

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

Apply Integral Reduction

= \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) ∣

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

Substitute back

u = arctan (\frac{4}{2} x)

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

Simplify

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

= x 4 x^{2} + 1 + \frac{1}{2} ln 2 x + 1 + 4 x^{2}

Add a constant to the solution

= x 4 x^{2} + 1 + \frac{1}{2} ln 2 x + 1 + 4 x^{2} + C

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What is the integral of sqrt(4+16x^2) ?
The integral of sqrt(4+16x^2) is xsqrt(4x^2+1)+1/2 ln|2x+sqrt(1+4x^2)|+C