Note: You can get full credit for this problem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit.

Question

Note: You can get full credit for this problem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit.

Consider the definite integral from pi/6 to pi/2 cos(z)/sin5(z)dz

Then the most appropriate substitution to simplify this integral is
u =

Then dz=f(z)du where
f(z) =

After making the substitution and simplifying we obtain the integral abg(u)du where
g(u) =
a =
b =

This definite integral has value =

Help please!!!! can't understand !!

Answer 1

I assume you mean

∫cos(z)/sin^5(z) dz
If u = sin(z), then du = cos(z) dz and we have
∫u^-5 du
= -1/4 u^-4

Now the limits of integration become [1/2,1] because u = sin(z)

Now it's easy to evaluate:

(-1/4 * 1^-4)-(-1/4 * 2^4) = 15/4

Answer 2

To solve this problem, we need to simplify the definite integral by making a substitution. Let's go step by step.

Step 1: Determine the appropriate substitution
Looking at the integrand, we can see that we have cos(z)/sin^5(z). To simplify this, we can consider using the substitution u = sin(z). This choice is reasonable because it allows us to convert the trigonometric expression into a more manageable algebraic expression.

Step 2: Find the differential dz in terms of du
To find dz in terms of du, we need to differentiate both sides of the substitution equation u = sin(z) with respect to z. We obtain:
du/dz = cos(z)

Rearranging this equation, we can express dz in terms of du:
dz = du/cos(z)

Step 3: Express the integrand in terms of u
Now we need to express cos(z)/sin^5(z) in terms of u. By using the trigonometric identity sin^2(z) = 1 - cos^2(z), we have sin^4(z) = (sin^2(z))^2 = (1 - cos^2(z))^2.

So our integrand cos(z)/sin^5(z) can be rewritten as:
cos(z)/sin^5(z) = cos(z)/(sin(z))^5 = cos(z)/(1 - cos^2(z))^2

Using the substitution u = sin(z), we can replace cos(z) with sqrt(1 - u^2):
cos(z) = sqrt(1 - u^2)

Our integrand becomes:
sqrt(1 - u^2)/(1 - u^2)^2

Step 4: Simplify the integrand
To simplify the integrand further, we can factor out (1 - u^2)^(-3/2) from the numerator and denominator:
sqrt(1 - u^2)/(1 - u^2)^2 = (1 - u^2)^(-3/2)

Now, we have the integrand in the form abg(u)du, where g(u) = (1 - u^2)^(-3/2).

Step 5: Determine the values of a and b
To determine the values of a and b, we need to evaluate the integral of g(u)du over the given limits of integration.

The definite integral of g(u)du from pi/6 to pi/2 can be evaluated by using the antiderivative of (1 - u^2)^(-3/2), which is given by the integral:
∫(1 - u^2)^(-3/2)du

Evaluating this integral over the given limits of integration will give us the values of a and b.

Step 6: Evaluate the definite integral
Finally, we evaluate the definite integral to find its value. Unfortunately, the definite integral of g(u)du from pi/6 to pi/2 cannot be expressed in simple terms. It requires the use of numerical methods or software to obtain an approximate value.

Therefore, to find the exact value of the definite integral, we need to use numerical integration techniques such as the trapezoidal rule, Simpson's rule, or software such as Mathematica, MATLAB, or numerical integration libraries in programming languages like Python.

Alternatively, if your goal is to simply answer the last question correctly, you can leave it as the definite integral of g(u)du from pi/6 to pi/2 and indicate that it cannot be evaluated exactly.