Скачать или смотреть Integral of sin^2(x)/cos^4(x) (substitution)

Integral of sin^2(x)/cos^4(x) (substitution)
To solve ∫ sin²(x) / cos⁴(x) dx using substitution:

1. Use the trigonometric identity:
sin²(x) = 1 - cos²(x).

The integral becomes:
∫ (1 - cos²(x)) / cos⁴(x) dx
= ∫ (1 / cos⁴(x)) - (cos²(x) / cos⁴(x)) dx
= ∫ sec⁴(x) - sec²(x) dx.

2. Split the integral:
∫ sec⁴(x) dx - ∫ sec²(x) dx.

3. For the second integral, ∫ sec²(x) dx:
The integral of sec²(x) is simply:
∫ sec²(x) dx = tan(x) + C.

4. For the first integral, ∫ sec⁴(x) dx:
Use the identity sec²(x) = 1 + tan²(x):
∫ sec⁴(x) dx = ∫ sec²(x) * sec²(x) dx
= ∫ (1 + tan²(x)) sec²(x) dx.

Use substitution: Let u = tan(x), so du = sec²(x) dx. The integral becomes:
∫ (1 + u²) du = u + (u³ / 3) + C
= tan(x) + (tan³(x) / 3) + C.

5. Combine the results:
∫ sec⁴(x) dx - ∫ sec²(x) dx
= tan(x) + (tan³(x) / 3) - tan(x) + C
= (tan³(x) / 3) + C.

Final answer:
∫ sin²(x) / cos⁴(x) dx = (tan³(x) / 3) + C.





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