Given that y=(x^3-2)^5 , find (dy)/(dx) b Hence , evaluate int _(0)^1x^2(x^3-2)^4dx __

Question

Answer 1

To find the derivative of \( y = (x^3 - 2)^5 \), we can use the chain rule. Let

, then

.The derivative of

with respect to

is:

First, compute

:

Next, compute

:

Now, combine these results:

So, the derivative is:

Next, we need to evaluate the integral:

We can use the substitution method. Let

, then

, or

.Change the limits of integration:- When

,

- When

,

Rewrite the integral in terms of

:

Evaluate the integral:

Calculate the definite integral:

So, the value of the integral is:

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