What is the integral of (5(1-tan^2(x)))/(sec^2(x)) ?

The integral of (5(1-tan^2(x)))/(sec^2(x)) is 5/2 sin(2x)+C

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integral of (5(1-tan^2(x)))/(sec^2(x))

\int \frac{5 ( 1 - tan ^{2} ( x ) )}{sec ^{2} ( x )} d x

\frac{5}{2} sin (2 x) + C

Solution steps

\int \frac{5 ( 1 - tan ^{2} ( x ) )}{sec ^{2} ( x )} d x

Take the constant out:

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

= 5 \cdot \int \frac{1 - tan ^{2} ( x )}{sec ^{2} ( x )} d x

Rewrite using trig identities

= 5 \cdot \int cos (2 x) d x

Apply u-substitution

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

= 5 \cdot \int cos (u) \frac{1}{2} d u

Take the constant out:

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

= 5 \cdot \frac{1}{2} \cdot \int cos (u) d u

Use the common integral:

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

= 5 \cdot \frac{1}{2} sin (u)

Substitute back

u = 2 x

= 5 \cdot \frac{1}{2} sin (2 x)

Simplify

5 \cdot \frac{1}{2} sin (2 x) : \frac{5}{2} sin (2 x)

= \frac{5}{2} sin (2 x)

Add a constant to the solution

= \frac{5}{2} sin (2 x) + C

x - 4

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What is the integral of (5(1-tan^2(x)))/(sec^2(x)) ?
The integral of (5(1-tan^2(x)))/(sec^2(x)) is 5/2 sin(2x)+C