What is the integral of 2sec^2(2x+y)tan(2x+y) ?

The integral of 2sec^2(2x+y)tan(2x+y) is 1/2 tan^2(y+2x)+C

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integral of 2sec^2(2x+y)tan(2x+y)

\int 2 sec^{2} (2 x + y) tan (2 x + y) d x

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

Solution steps

\int 2 sec^{2} (2 x + y) tan (2 x + y) d x

Take the constant out:

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

= 2 \cdot \int sec^{2} (y + 2 x) tan (y + 2 x) d x

Apply u-substitution

= 2 \cdot \int sec^{2} (u) tan (u) \frac{1}{2} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int sec^{2} (u) tan (u) d u

Apply u-substitution

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

Apply the Power Rule

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

Substitute back

= 2 \cdot \frac{1}{2} \cdot \frac{tan ^{2} ( y + 2 x )}{2}

Simplify

2 \cdot \frac{1}{2} \cdot \frac{tan ^{2} ( y + 2 x )}{2} : \frac{1}{2} tan^{2} (y + 2 x)

= \frac{1}{2} tan^{2} (y + 2 x)

Add a constant to the solution

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

What is the integral of 2sec^2(2x+y)tan(2x+y) ?
The integral of 2sec^2(2x+y)tan(2x+y) is 1/2 tan^2(y+2x)+C