Many formulas There are several ways to express the indefinite...

Question

Many formulas There are several ways to express the indefinite integral of sech x.

b. Show that ∫ sech x dx = sin⁻¹ (tanh x) + C. (Hint: Show that sech x = sech² x / √(1 − tanh² x) and then make a change of variables.)

Accepted Answer

Welcome back, everyone. Compute the indefinite integral se of X divided by 2D X. For this problem we're going to begin with the identity. squared of x divided by 2. Equals 1 minus. 10 square of X divided by 2. Now if we solve for sash, we can show that sash of x divided by 2 is equal to square root of the right hand side. Note that we will only take the positive solution because sesh of x divided by 2 is. Greater Damn, 0. For all excess. So se of X divided by 2 can be written as square root of 1 minus. in squared of x divided by 2. They are for sash. Of X divided by 2. Can also be written in the form of squared of x divided by 2. Divided by se of X divided by 2, right? Because a square of a number divided by that number, as long as the number is positive, is equal to the original number itself. So we can now write squared of X divided by 2 in the numerator. And in the denominator we can use square root of. 1 minus tinge squared of x divided by 2. And when we integrate, we can actually use the use substitution. So our integral of of X divided by 2D X can first be written as the integral of squared of x divided by 2 divided by. Square root of 1 minus. Ange Squared Of X divided by 2, followed by D X. Now let's apply you substitution. Let's suppose that you equals. 10 of X divided by 2. We can then show that DU is equal to 1/2 squared of x divided by 2 D X. Using the chain rule. And now let's apply this substitution within our integral. We get integral of. Our numerator contains squared of x divided by 2D X, which is simply DU divided by 12, and that's 2DU. Divided by square root of 1 minus squared off x divided by 2. So that's 1 minus U squared. We can factor out the constant, we get to integral of the EU divided by. Square root of 1 minus u squared and this integral represents the inverse of sine so we get two inverse of sine. Of tinge. Of X divided by 2 + a constant of integration C. That's our final answer for this problem, and thank you for watching.

Many formulas There are several ways to express the indefinite integral of sech x.

b. Show that ∫ sech x dx = sin⁻¹ (tanh x) + C. (Hint: Show that sech x = sech² x / √(1 − tanh² x) and then make a change of variables.)

Key Concepts

Hyperbolic Functions and Their Identities

Substitution Method in Integration

Inverse Hyperbolic and Trigonometric Functions

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