Problem 11
Question
Find the integrals. $$ \int \frac{y}{\sqrt{5-y}} d y $$
Step-by-Step Solution
Verified Answer
The integral is
\( -(10\sqrt{5-y} - \frac{2}{3}(5-y)^{3/2}) + C \).
1Step 1: Identify the Substitution
To solve the integral \( \int \frac{y}{\sqrt{5-y}} \ dy \), we can use substitution. We notice that the expression under the square root can be simplified if we let \( u = 5 - y \). This substitution will help us simplify the integral more easily.
2Step 2: Substitute and Express in Terms of Variable u
With the substitution \( u = 5 - y \), we find the differential \( du = -dy \), or \( dy = -du \). We substitute these into the integral to change the variable:\[\int \frac{y}{\sqrt{5-y}} \, dy = -\int \frac{(5-u)}{\sqrt{u}} \, du\]
3Step 3: Simplify the Integral
Now, simplify the integral:\[-\int \frac{(5-u)}{\sqrt{u}} \, du = -\left( \int \frac{5}{\sqrt{u}} \, du - \int \frac{u}{\sqrt{u}} \, du \right)\]This can be rewritten as:\[-\left( 5 \int u^{-1/2} \, du - \int u^{1/2} \, du \right)\]
4Step 4: Integrate Simplified Expressions
Integrate each term separately:1. \( 5 \int u^{-1/2} \, du = 5 \cdot 2u^{1/2} = 10\sqrt{u} \)2. \( \int u^{1/2} \, du = \frac{2}{3}u^{3/2} \)Combine these results:\[-(10\sqrt{u} - \frac{2}{3}u^{3/2})\]
5Step 5: Back Substitute in Terms of y
Replace \( u \) with \( 5 - y \):\[-(10\sqrt{5-y} - \frac{2}{3}(5-y)^{3/2}) + C\]Where \( C \) is the constant of integration.
Key Concepts
Substitution MethodSquare Root IntegrationBack Substitution
Substitution Method
The substitution method is a valuable technique in integral calculus, aiding in transforming complex integrals into more manageable forms. In the problem, we aim to solve the integral \( \int \frac{y}{\sqrt{5-y}} \, dy \). Often, this method involves recognizing parts of the integrand that can be replaced with simpler expressions.
This can frequently occur when you spot a composition of functions, such as variables inside a square root or exponent, which can be rewritten through substitution.
To apply this, you identify a substitution that simplifies the integral. Here, we substitute \( u = 5 - y \). This transforms our unfamiliar integral into a more familiar expression in terms of \( u \).
Some key steps include:
This can frequently occur when you spot a composition of functions, such as variables inside a square root or exponent, which can be rewritten through substitution.
To apply this, you identify a substitution that simplifies the integral. Here, we substitute \( u = 5 - y \). This transforms our unfamiliar integral into a more familiar expression in terms of \( u \).
Some key steps include:
- Choose a substitution based on part of the integrand.
- Calculate the differential of \( u \), here \( du = -dy \).
- Rewrite the integral in terms of \( u \) and \( du \).
Square Root Integration
Integrating expressions involving square roots can initially seem daunting due to their complexity. However, they can become manageable with substitution.
In our exercise, after the substitution \( u = 5 - y \), it simplified our integral to deal with square roots more easily. Square root integration essentially focuses on expressions like \( \sqrt{u} \) and demands careful separation to achieve solvable results.
For this exercise, the integral \( \int \frac{(5-u)}{\sqrt{u}} \, du \) is separated into two distinct parts:
In our exercise, after the substitution \( u = 5 - y \), it simplified our integral to deal with square roots more easily. Square root integration essentially focuses on expressions like \( \sqrt{u} \) and demands careful separation to achieve solvable results.
For this exercise, the integral \( \int \frac{(5-u)}{\sqrt{u}} \, du \) is separated into two distinct parts:
- \( \int \frac{5}{\sqrt{u}} \, du \)
- \( \int \frac{u}{\sqrt{u}} \, du \)
Back Substitution
The final essential step in substitution problems is back substitution. After solving the integral in terms of \( u \), it becomes necessary to revert to the original variable to reach the solution appropriate for the problem.
Back substitution involves replacing \( u \) with its original expression, which was \( u = 5 - y \) in our exercise.
This replacement converts the solution expressed in terms of \( u \) back to terms of \( y \), which is important because it aligns the solution with the original problem statement.
Incorporating back substitution is done as follows:
Back substitution involves replacing \( u \) with its original expression, which was \( u = 5 - y \) in our exercise.
This replacement converts the solution expressed in terms of \( u \) back to terms of \( y \), which is important because it aligns the solution with the original problem statement.
Incorporating back substitution is done as follows:
- Take the evaluated integral in terms of \( u \): \(-(10\sqrt{u} - \frac{2}{3}u^{3/2}) + C\)
- Replace \( u \) with \( 5 - y \), giving: \(-(10\sqrt{5-y} - \frac{2}{3}(5-y)^{3/2}) + C\)
Other exercises in this chapter
Problem 10
Find the integrals in problems. Check your answers by differentiation. $$ \int x^{2}\left(1+2 x^{3}\right)^{2} d x $$
View solution Problem 10
Find an antiderivative. $$ k(x)=10+8 x^{3} $$
View solution Problem 11
Using the Fundamental Theorem, evaluate the definite integrals in problem exactly. $$ \int_{4}^{9} \sqrt{x} d x $$
View solution Problem 11
Find the integrals in problems. Check your answers by differentiation. $$ \int x\left(x^{2}-4\right)^{7 / 2} d x $$
View solution