Mastering The Integral Of Sec^2 X: Proven Steps To Obtain The Solution Of ∫Sec^2 X Dx = Tan X + C.

sec^2 dx

tanx + C

The integral of sec^2 x dx is:

∫sec^2 x dx = tan x + C

Where C is the constant of integration.

To prove this, we need to use a trigonometric identity. The identity states that:

1 + tan^2 x = sec^2 x

Rearranging the equation, we get:

tan^2 x = sec^2 x – 1

Dividing both sides of the equation by sec^2 x, we get:

tan^2 x / sec^2 x = sec^2 x / sec^2 x – 1 / sec^2 x

tan^2 x / sec^2 x = 1 – cos^2 x / sin^2 x

Since tan x = sin x / cos x, we can substitute and get:

sin^2 x / cos^2 x / sec^2 x = 1 – cos^2 x / sin^2 x

Simplifying the left side:

sin^2 x / cos^2 x * cos^2 x / 1 = sin^2 x

Hence:

sin^2 x = sin^2 x

This proves the identity that we will use to obtain the integral of sec^2 x dx:

∫sec^2 x dx = ∫(1 + tan^2 x) dx

Using the power rule of integration, we get:

∫sec^2 x dx = tan x + C

Where C is the constant of integration.

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