What is the integral of ((θ+1))/(sqrt(θ^2+2θ+4)) ?

The integral of ((θ+1))/(sqrt(θ^2+2θ+4)) is sqrt(θ^2+2θ+4)+C

integral of ((θ+1))/(sqrt(θ^2+2θ+4))

\int \frac{( θ + 1 )}{θ ^{2} + 2 θ + 4} d θ

θ^{2} + 2 θ + 4 + C

Solution steps

\int \frac{θ + 1}{θ ^{2} + 2 θ + 4} d θ

Apply u-substitution

= \int \frac{1}{2 u} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

Substitute back

u = θ^{2} + 2 θ + 4

= \frac{1}{2} \cdot 2 (θ^{2} + 2 θ + 4)^{\frac{1}{2}}

Simplify

\frac{1}{2} \cdot 2 (θ^{2} + 2 θ + 4)^{\frac{1}{2}} : θ^{2} + 2 θ + 4

= θ^{2} + 2 θ + 4

Add a constant to the solution

= θ^{2} + 2 θ + 4 + C

\int cos (x + 5) d x

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What is the integral of ((θ+1))/(sqrt(θ^2+2θ+4)) ?
The integral of ((θ+1))/(sqrt(θ^2+2θ+4)) is sqrt(θ^2+2θ+4)+C