What is the integral of (5e(5sin(4x)))/(sec(4x)) ?

The integral of (5e(5sin(4x)))/(sec(4x)) is -(25e)/8 cos^2(4x)+C

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integral of (5e(5sin(4x)))/(sec(4x))

\int \frac{5 e ( 5 sin ( 4 x ) )}{sec ( 4 x )} d x

- \frac{25 e}{8} cos^{2} (4 x) + C

Solution steps

\int \frac{5 e 5 sin ( 4 x )}{sec ( 4 x )} d x

Take the constant out:

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

= 5 e 5 \cdot \int \frac{sin ( 4 x )}{sec ( 4 x )} d x

Simplify

= 25 e \cdot \int \frac{sin ( 4 x )}{sec ( 4 x )} d x

Use the basic trigonometric identity:

\frac{1}{sec ( x )} = cos (x)

= 25 e \cdot \int sin (4 x) cos (4 x) d x

Apply u-substitution

: \int - \frac{u}{4} d u

= 25 e \cdot \int - \frac{u}{4} d u

Take the constant out:

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

= 25 e (- \frac{1}{4} \cdot \int u d u)

Apply the Power Rule

: \frac{u ^{2}}{2}

= 25 e (- \frac{1}{4} \cdot \frac{u ^{2}}{2})

Substitute back

u = cos (4 x)

= 25 e (- \frac{1}{4} \cdot \frac{cos ^{2} ( 4 x )}{2})

Simplify

25 e (- \frac{1}{4} \cdot \frac{cos ^{2} ( 4 x )}{2}) : - \frac{25 e}{8} cos^{2} (4 x)

= - \frac{25 e}{8} cos^{2} (4 x)

Add a constant to the solution

= - \frac{25 e}{8} cos^{2} (4 x) + C

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What is the integral of (5e(5sin(4x)))/(sec(4x)) ?
The integral of (5e(5sin(4x)))/(sec(4x)) is -(25e)/8 cos^2(4x)+C