Question

Give the Result please
$\huge \red \int{ \blue{\sec(x) { \green\times } \tan(x) }}$
​

Answer 1

EXPLANATION.

⇒ ∫[sec(x).tan(x)]dx.

As we know that,

Formula of :

⇒ ∫sec x tan x dx = sec x + c.

Using this formula in equation, we get.

⇒ ∫[sec(x).tan(x)]dx = sec(x) + c.

Proof :

⇒ ∫[sec(x).tan(x)]dx.

⇒ ∫1/cos(x).sin(x)/cos(x) dx.

⇒ ∫sin(x)dx/cos²x.

By using substitution method, we get.

Let we assume that,

⇒ cos(x) = t.

Differentiate w.r.t x, we get.

⇒ - sin(x)dx = dt.

Using this formula in equation, we get.

⇒ ∫- dt/t² = -(-1/t) + c.

⇒ 1/t + c.

Put the value of t = cos(x) in equation, we get.

⇒ 1/cos(x) + c = sec(x) + c.

                                                                                                                               

MORE INFORMATION.

Standard integrals.

(1) = ∫0.dx = c.

(2) = ∫1.dx = x + c.

(3) = ∫k dx = kx + c, (k ∈ R).

(4) = ∫xⁿdx = xⁿ⁺¹/n + 1 + c, (n ≠ - 1).

(5) = ∫dx/x = ㏒(x) + c.

(6) = ∫eˣdx = eˣ + c.

(7) = ∫aˣdx = aˣ/㏒(a) + c = aˣ㏒(e) + c.

Answer 2

Solution:

$\bf \int \: sec(x) \times \tan(x)$

We can integrate it easily,

Now we can you substitution method,

Since the derivative of sec (x) is

$\bf→ sec (x) × tan (x),$

Now the integral of,

$\bf→ sec (x) × tan (x) \: is \: sec (x)$

Then the answer is,

$\bf→ sec (x) + C$

Final answer:

$\bf sec (x) + C$ is the final result.