What is the integral of 1/(2sin^2(x)+cos^2(x)) ?

The integral of 1/(2sin^2(x)+cos^2(x)) is 1/(sqrt(2))arctan(sqrt(2)tan(x))+C

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integral of 1/(2sin^2(x)+cos^2(x))

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

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

Solution steps

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

Rewrite using trig identities

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

Apply Trigonometric Substitution

: \int \frac{1}{1 + 2 u ^{2}} d u

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

Apply Integral Substitution

= \int \frac{1}{2 ( 1 + v ^{2} )} d v

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}{1 + v ^{2}} d v

Use the common integral:

\int \frac{1}{1 + v ^{2}} d v = arctan (v)

= \frac{1}{2} arctan (v)

Substitute back

= \frac{1}{2} arctan (2 tan (x))

Add a constant to the solution

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

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What is the integral of 1/(2sin^2(x)+cos^2(x)) ?
The integral of 1/(2sin^2(x)+cos^2(x)) is 1/(sqrt(2))arctan(sqrt(2)tan(x))+C