integral of sec²x
The integral of sec²(x) can be found by using a technique called integration by substitution
The integral of sec²(x) can be found by using a technique called integration by substitution. Here’s how:
Let’s start by using a trigonometric identity to rewrite sec²(x). The identity we will use is:
sec²(x) = 1 + tan²(x)
Now, let’s rewrite the integral:
∫sec²(x) dx = ∫(1 + tan²(x)) dx
Now, we can split this integral into two separate integrals:
∫(1 + tan²(x)) dx = ∫1 dx + ∫tan²(x) dx
The first integral, ∫1 dx, is simply the integral of a constant, which is x:
∫1 dx = x + C1 (C1 is the constant of integration)
The second integral, ∫tan²(x) dx, requires the use of a trigonometric identity. The identity we will use is:
tan²(x) = sec²(x) – 1
Now, let’s rewrite the integral again:
∫tan²(x) dx = ∫(sec²(x) – 1) dx
∫tan²(x) dx = ∫sec²(x) dx – ∫1 dx
Substituting back the original integral, we have:
∫tan²(x) dx = ∫sec²(x) dx – x
Now, we can solve for the integral of sec²(x):
∫sec²(x) dx = ∫tan²(x) dx + x
Finally, using a trigonometric identity, we can simplify the integral of tan²(x):
Using the identity tan²(x) = sec²(x) – 1:
∫sec²(x) dx = ∫(sec²(x) – 1) dx + x
∫sec²(x) dx = ∫sec²(x) dx – ∫1 dx + x
Rearranging the equation, we get:
0 = – ∫1 dx + x
∫1 dx = x + C2 (C2 is another constant of integration)
Therefore, the integral of sec²(x) is:
∫sec²(x) dx = tan(x) + x + C (where C = C1 + C2)
So, the final answer is:
∫sec²(x) dx = tan(x) + x + C, where C is the constant of integration.
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