283. $$ \int \frac{y^{5}}{\left(1-y^{3}\right)^{3 / 2}} d y $$

Question

UpStudy AI · Accepted Answer

To solve the integral

$$
\int \frac{y^{5}}{(1-y^{3})^{3/2}} \, dy,
$$

we can use a substitution method. Let's proceed step by step.

### Step 1: Substitution
We can let

$$
u = 1 - y^3.
$$

Then, we differentiate $$u$$ with respect to $$y$$:

$$
\frac{du}{dy} = -3y^2 \implies du = -3y^2 \, dy \implies dy = -\frac{du}{3y^2}.
$$

### Step 2: Express $$y^5$$ in terms of $$u$$
From our substitution, we have:

$$
y^3 = 1 - u \implies y = (1 - u)^{1/3}.
$$

Thus,

$$
y^5 = (y^3)^{5/3} = (1 - u)^{5/3}.
$$

### Step 3: Substitute into the integral
Now we can substitute $$y^5$$ and $$dy$$ into the integral:

$$
\int \frac{(1-u)^{5/3}}{(u)^{3/2}} \left(-\frac{du}{3y^2}\right).
$$

We also need to express $$y^2$$ in terms of $$u$$:

$$
y^2 = (1 - u)^{2/3}.
$$

Thus,

$$
dy = -\frac{du}{3(1-u)^{2/3}}.
$$

### Step 4: Rewrite the integral
Now we can rewrite the integral:

$$
-\frac{1}{3} \int \frac{(1-u)^{5/3}}{u^{3/2} (1-u)^{2/3}} \, du = -\frac{1}{3} \int \frac{(1-u)^{3/3}}{u^{3/2}} \, du = -\frac{1}{3} \int \frac{1-u}{u^{3/2}} \, du.
$$

### Step 5: Split the integral
This can be split into two separate integrals:

$$
-\frac{1}{3} \left( \int u^{-3/2} \, du - \int u^{-1/2} \, du \right).
$$

### Step 6: Solve the integrals
Now we can solve each integral:

1. $$\int u^{-3/2} \, du = -2u^{-1/2} + C_1$$
2. $$\int u^{-1/2} \, du = 2u^{1/2} + C_2$$

Putting it all together:

$$
-\frac{1}{3} \left( -2u^{-1/2} - 2u^{1/2} \right) = \frac{2}{3} u^{-1/2} - \frac{2}{3} u^{1/2} + C.
$$

### Step 7: Substitute back for $$u$$
Now we substitute back $$u = 1 - y^3$$:

$$
\frac{2}{3} (1 - y^3)^{-1/2} - \frac{2}{3} (1 - y^3)^{1/2} + C.
$$

### Final Answer
Thus, the integral evaluates to:

$$
\int \frac{y^{5}}{(1-y^{3})^{3/2}} \, dy = \frac{2}{3} (1 - y^3)^{-1/2} - \frac{2}{3} (1 - y^3)^{1/2} + C.
$$
The integral evaluates to: $$ \frac{2}{3} (1 - y^3)^{-1/2} - \frac{2}{3} (1 - y^3)^{1/2} + C. $$

Solución de inteligencia artificial de Upstudy

Responder

Solución

Step 1: Substitution

Step 2: Express in terms of

Step 3: Substitute into the integral

Step 4: Rewrite the integral

Step 5: Split the integral

Step 6: Solve the integrals

Step 7: Substitute back for

Final Answer

The Deep Dive

preguntas relacionadas

Latest Calculus Questions