Question

Y=cos^4(3x+7^5x-3)+(4x^3-5x)×ln(2x+1)

Answer 1

Что надо сделать
я что-то не пончл

Answer 2

To simplify the expression Y = cos^4(3x + 7^5x - 3) + (4x^3 - 5x) × ln(2x + 1), we can break it down step by step:

Step 1: Simplify the trigonometric expression. Y = cos^4(3x + 7^5x - 3)

Step 2: Simplify the logarithmic expression. Y = (4x^3 - 5x) × ln(2x + 1)

Let's first work on Step 1:

Step 1: Simplify the trigonometric expression. Y = cos^4(3x + 7^5x - 3)

Now, we need to use trigonometric identities to simplify this expression. The trigonometric identity we'll use is:

cos^2(x) = (1 + cos(2x)) / 2

Using this identity twice, we get:

cos^4(x) = [cos^2(x)]^2 = [(1 + cos(2x)) / 2]^2 = (1 + cos(2x))^2 / 4 = (1 + 2cos(2x) + cos^2(2x)) / 4

Now, we need to find cos(2x) in terms of cos(x):

cos(2x) = cos(x + x) = cos^2(x) - sin^2(x) = cos^2(x) - (1 - cos^2(x)) = 2cos^2(x) - 1

Substitute this value back into the expression:

cos^4(x) = (1 + 2cos(2x) + cos^2(2x)) / 4 = (1 + 2(2cos^2(x) - 1) + (2cos^2(x) - 1)^2) / 4 = (1 + 4cos^2(x) - 2 + 4cos^4(x) - 4cos^2(x) + 1) / 4 = (4cos^4(x) + 1) / 4 = cos^4(x) + 1/4

Now, apply this to our original expression:

Y = cos^4(3x + 7^5x - 3) ≈ cos^4(3x) + 1/4

Next, let's work on Step 2:

Step 2: Simplify the logarithmic expression. Y = (4x^3 - 5x) × ln(2x + 1)

There's not much we can do to simplify the logarithmic expression further, so the final simplified expression is:

Y ≈ cos^4(3x) + 1/4 + (4x^3 - 5x) × ln(2x + 1)

Please note that this is an approximate simplification since we've used some trigonometric identities and made some assumptions. If you need a more precise evaluation for a specific value of x, you can substitute that value into the expression.