What is the integral of 8/(3cos(2x)+1) ?

The integral of 8/(3cos(2x)+1) is sqrt(2)ln| 1/(sqrt(2))tan(x)+1|-sqrt(2)ln| 1/(sqrt(2))tan(x)-1|+C

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integral of 8/(3cos(2x)+1)

\int \frac{8}{3 cos ( 2 x ) + 1} d x

2 ln \frac{1}{2} tan (x) + 1 - 2 ln \frac{1}{2} tan (x) - 1 + C

Solution steps

\int \frac{8}{3 cos ( 2 x ) + 1} d x

Take the constant out:

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

= 8 \cdot \int \frac{1}{3 cos ( 2 x ) + 1} d x

Apply u-substitution

: \int \frac{1}{2 ( 3 cos ( u ) + 1 )} d u

= 8 \cdot \int \frac{1}{2 ( 3 cos ( u ) + 1 )} d u

Take the constant out:

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

= 8 \cdot \frac{1}{2} \cdot \int \frac{1}{3 cos ( u ) + 1} d u

Apply u-substitution

: \int \frac{1}{- v ^{2} + 2} d v

= 8 \cdot \frac{1}{2} \cdot \int \frac{1}{- v ^{2} + 2} d v

Apply Integral Substitution

= 8 \cdot \frac{1}{2} \cdot \int \frac{1}{2 ( - w ^{2} + 1 )} d w

Take the constant out:

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

= 8 \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \int \frac{1}{- w ^{2} + 1} d w

Use the common integral:

\int \frac{1}{- w ^{2} + 1} d w = \frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}

= 8 \cdot \frac{1}{2} \cdot \frac{1}{2} (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2})

Substitute back

= 8 \cdot \frac{1}{2} \cdot \frac{1}{2} \frac{ln \frac{t a n ( \frac{2 x}{2} )}{2} + 1}{2} - \frac{ln \frac{t a n ( \frac{2 x}{2} )}{2} - 1}{2}

Simplify

8 \cdot \frac{1}{2} \cdot \frac{1}{2} \frac{ln \frac{t a n ( \frac{2 x}{2} )}{2} + 1}{2} - \frac{ln \frac{t a n ( \frac{2 x}{2} )}{2} - 1}{2} : 2 ln \frac{1}{2} tan (x) + 1 - 2 ln \frac{1}{2} tan (x) - 1

= 2 ln \frac{1}{2} tan (x) + 1 - 2 ln \frac{1}{2} tan (x) - 1

Add a constant to the solution

= 2 ln \frac{1}{2} tan (x) + 1 - 2 ln \frac{1}{2} tan (x) - 1 + C

x \to 2 + lim (\frac{1}{x ^{2} - 4})

\int \frac{ln ( sin ( x ) )}{tan ( x )} d x

\frac{x}{9} - \frac{9}{x}

f (x) = e^{c o s (x)}

x^{2} + 2 y^{2} = 3, (1, 1)

What is the integral of 8/(3cos(2x)+1) ?
The integral of 8/(3cos(2x)+1) is sqrt(2)ln| 1/(sqrt(2))tan(x)+1|-sqrt(2)ln| 1/(sqrt(2))tan(x)-1|+C