What is the integral of (sin(2θ))/(sqrt(sin^2(θ)+1)) ?

The integral of (sin(2θ))/(sqrt(sin^2(θ)+1)) is 2sqrt(sin^2(θ)+1)+C

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integral of (sin(2θ))/(sqrt(sin^2(θ)+1))

\int \frac{sin ( 2 θ )}{sin ^{2} ( θ ) + 1} d θ

2 sin^{2} (θ) + 1 + C

Solution steps

\int \frac{sin ( 2 θ )}{sin ^{2} ( θ ) + 1} d θ

Apply u-substitution

: \int \frac{1}{u} d u

= \int \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= 2 u^{\frac{1}{2}}

Substitute back

u = sin^{2} (θ) + 1

= 2 (sin^{2} (θ) + 1)^{\frac{1}{2}}

Apply radical rule:

a^{\frac{1}{n}} = n a

= 2 sin^{2} (θ) + 1

Add a constant to the solution

= 2 sin^{2} (θ) + 1 + C

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What is the integral of (sin(2θ))/(sqrt(sin^2(θ)+1)) ?
The integral of (sin(2θ))/(sqrt(sin^2(θ)+1)) is 2sqrt(sin^2(θ)+1)+C