Question

S(sin 8x+7tg2x-12x³)dx​

Answer 1

sin 8x+9tgx²-12x⁵dn....

Answer 2

Ответ:

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Answer 3

To integrate the given expression ∫S(sin(8x + 7tan^2(x) - 12x^3) dx, we need to simplify the expression inside the sine function and then integrate term by term. Let's break it down step by step:

1. Start with the innermost function: 8x + 7tan^2(x) - 12x^3

2. We can simplify tan^2(x) using the trigonometric identity: tan^2(x) = sec^2(x) - 1

   So, our expression becomes: 8x + 7(sec^2(x) - 1) - 12x^3

3. Distribute the constants: 8x + 7sec^2(x) - 7 - 12x^3

4. Now, rewrite sec^2(x) in terms of sin^2(x): sec^2(x) = 1 + tan^2(x)

   So, we have: 8x + 7(1 + tan^2(x)) - 7 - 12x^3

5. Distribute the constants again: 8x + 7 + 7tan^2(x) - 7 - 12x^3

6. Simplify further: 8x - 12x^3 + 7tan^2(x)

Now, our integral expression becomes ∫S(sin(8x - 12x^3 + 7tan^2(x)) dx.

To integrate this expression, you may need to use numerical or computational methods, as it involves a polynomial term (12x^3) and a trigonometric term (sin) with a complicated argument. There might not be a simple closed-form expression for this integral. You can use software or calculators to approximate the result if needed.