Solving the Integral of sec(u) du using Substitution Method.

∫ secu du

To solve the integral of sec(u) du, we can use a substitution method

To solve the integral of sec(u) du, we can use a substitution method. Let’s use the substitution u = cos(x). Then, du = -sin(x) dx.

Now, we need to express sec(u) in terms of u. Note that sec(u) is the reciprocal of cos(u). Since we have u = cos(x), we can use the identity sec(u) = 1/cos(u) to get sec(u) = 1/cos(cos(x)).

Substituting these values back into our integral, we have ∫ sec(u) du = ∫ (1/cos(cos(x))) (-sin(x)) dx.

Now, we have transformed our integral into an integral in terms of x. We can rewrite it as ∫ (1/cos(cos(x))) (-sin(x)) dx.

To simplify further, we can use the identity cos(cos(x)) = cos²(x) – sin²(x) = 1 – sin²(x) – sin²(x) = 1 – 2sin²(x). Substituting this into the integral, we have ∫ (1/cos(cos(x))) (-sin(x)) dx = ∫ (1/(1 – 2sin²(x))) (-sin(x)) dx.

Now, we can apply another substitution to simplify our integral. Let’s use u = sin(x), which gives us du = cos(x) dx. Therefore, dx = du/cos(x) = du/sqrt(1 – u²).

Substituting these new values into our integral, we have ∫ (1/(1 – 2sin²(x))) (-sin(x)) dx = ∫ (1/(1 – 2u²)) (-du/sqrt(1 – u²)).

Finally, our integral becomes ∫ (1/(1 – 2u²)) (-du/sqrt(1 – u²)).

This integral can be solved using trigonometric substitution or partial fractions, depending on the specific form of the integrand. However, without further information about the limits of integration or the desired form of the solution, it is not possible to provide an exact numerical result.

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