What is the integral of sqrt(2)asqrt(1+cos(θ)) ?

The integral of sqrt(2)asqrt(1+cos(θ)) is 4atan(θ/2)cos(θ/2)+C

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integral of sqrt(2)asqrt(1+cos(θ))

\int 2 a 1 + cos (θ) d θ

4 a tan (\frac{θ}{2}) cos (\frac{θ}{2}) + C

Solution steps

\int 2 a 1 + cos (θ) d θ

Take the constant out:

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

= 2 a \cdot \int 1 + cos (θ) d θ

Apply u-substitution

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

Take the constant out:

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

= 2 a \cdot 22 \cdot \int \frac{1}{( 1 + u ^{2} ) 1 + u ^{2}} d u

Simplify

(1 + u^{2}) 1 + u^{2} : (1 + u^{2})^{\frac{3}{2}}

= 2 a \cdot 22 \cdot \int \frac{1}{( 1 + u ^{2} ) ^{\frac{3}{2}}} d u

Apply Trigonometric Substitution

= 2 a \cdot 22 \cdot \int cos (v) d v

Use the common integral:

\int cos (v) d v = sin (v)

= 2 a \cdot 22 sin (v)

Substitute back

= 2 a \cdot 22 sin (arctan (tan (\frac{θ}{2})))

Simplify

2 a \cdot 22 sin (arctan (tan (\frac{θ}{2}))) : 4 a tan (\frac{θ}{2}) cos (\frac{θ}{2})

= 4 a tan (\frac{θ}{2}) cos (\frac{θ}{2})

Add a constant to the solution

= 4 a tan (\frac{θ}{2}) cos (\frac{θ}{2}) + C

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