Evaluating the Integral of sec^2x tanx: Techniques and Variations

Understanding how to evaluate the integral of sec^2x tanx is crucial for mastering integration techniques in calculus. This article presents various methods, including substitution, integration by parts, and a straightforward approach. We'll also explore why seemingly different results are actually equivalent and how to recognize and apply these techniques.

Method 1: Substitution

One of the most straightforward ways to evaluate the integral of sec^2x tanx is through substitution. The integral can be rewritten as:

int sec^2x tanx dx int secx (secx tanx) dx

Let u secx, then du secx tanx dx. Substituting these into the integral, we get:

int u du frac{1}{2}u^2 C frac{1}{2}sec^2x C

Alternatively, if we let u tanx, then du sec^2x dx. This gives us:

int u du frac{1}{2}u^2 C frac{1}{2}tan^2x C

These results appear different but are actually equivalent, as we'll show in the next section.

Method 2: Integration by Parts

Integration by parts is another method to evaluate the integral. We can use the formulas:

u tanx, dv sec^2x dx

Then, du sec^2x dx, v tanx. Applying integration by parts:

int u dv uv - int v du

Substituting these values:

int sec^2x tanx dx tan^2x - int sec^2x tanx dx

Adding int sec^2x tanx dx to both sides:

2 int sec^2x tanx dx tan^2x C

Thus, the integral is:

int sec^2x tanx dx frac{1}{2}tan^2x C

Method 3: Direct Evaluation Using Trigonometric Identities

Another approach is to directly evaluate the integral using trigonometric identities. Recall that the derivative of tanx is sec^2x. Thus:

int sec^2x tanx dx int tanx sec^2x dx

This can be written as:

int tanx sec^2x dx int x dx frac{x^2}{2} C frac{tan^2x}{2} C

Note that this result is the same as the ones obtained from the previous methods, confirming the equivalence of these results.

Why Different Results Are Equivalent

The results obtained from the substitution methods differ by a constant. For instance, the integral can be expressed as:

frac{1}{2}tan^2x C_1 frac{1}{2}sec^2x - 1 C_1 frac{1}{2}sec^2x C_2

This shows that the two forms are equivalent, differing only by a constant. This is consistent with the fundamental theorem of calculus, which states that any two antiderivatives of a function differ by a constant.

Conclusion

The integral of sec^2x tanx can be evaluated using various techniques, including substitution, integration by parts, and direct evaluation using trigonometric identities. These methods all lead to the same result, confirming the equivalence of the results. As you practice these techniques, you'll become more adept at recognizing when and how to apply them, making the process of integration more straightforward.