integration by parts:
1. e^(sqrt(3x+9))
2. sin(lr)
are the answers:
1. 2e^(sqrt(3x+9))(sqrt(3x+9)-1) + C
2. -r cos(lnr) + sin(lr) + C
To integrate the given expressions using integration by parts, you need to follow a specific method. Here's a step-by-step explanation of how to solve each problem:
1. Integration of e^(sqrt(3x+9)):
Step 1: Identify the function to be differentiated and the function to be integrated.
In this case, let's assign u = e^(sqrt(3x+9)) and dv = dx.
Step 2: Calculate du (the derivative of u) and v (the integral of dv).
Differentiating u with respect to x will give us du. In this case, du = (1/2)e^(sqrt(3x+9))/(sqrt(3x+9)) dx.
Integrating dv, which is just dx, gives us v = x.
Step 3: Apply the integration by parts formula.
The integration by parts formula is as follows:
∫ u dv = uv - ∫ v du
Using this formula, we can rewrite the given expression as follows:
∫ e^(sqrt(3x+9)) dx = uv - ∫ v du
= (e^(sqrt(3x+9)) * x) - ∫ (x * (1/2)e^(sqrt(3x+9))/(sqrt(3x+9))) dx
= xe^(sqrt(3x+9)) - (1/2)∫ e^(sqrt(3x+9))/(sqrt(3x+9)) dx
Step 4: Simplify the remaining integral.
The remaining integral, ∫ e^(sqrt(3x+9))/(sqrt(3x+9)) dx, can be simplified by making a substitution. Let's substitute y = sqrt(3x+9):
dy/dx = (3/2)(1/sqrt(3x+9))
dy = (3/2)(1/sqrt(3x+9)) dx
dx = 2/3(sqrt(3x+9)) dy
Now we can rewrite the integral in terms of y:
(1/2)∫ e^y dy
This integral is simply e^y + C.
Step 5: Substitute back the original variable.
Substituting y = sqrt(3x+9) and e^y = e^(sqrt(3x+9)) into the previous step, we have:
(1/2)e^(sqrt(3x+9))
Step 6: Combine the simplified integral with the previous terms.
Returning to the previous expression, we have:
xe^(sqrt(3x+9)) - (1/2)e^(sqrt(3x+9)) + C
Simplifying, we get the final answer:
2e^(sqrt(3x+9))(sqrt(3x+9)-1) + C
2. Integration of sin(lr):
Step 1: Identify the function to be differentiated and the function to be integrated.
In this case, let's assign u = sin(lr) and dv = 1 dx.
Step 2: Calculate du (the derivative of u) and v (the integral of dv).
Differentiating u with respect to x will give us du. In this case, du = l cos(lr) dx.
Integrating dv, which is just 1 dx, gives us v = x.
Step 3: Apply the integration by parts formula.
Using the integration by parts formula, we can rewrite the given expression as follows:
∫ sin(lr) dx = uv - ∫ v du
= x sin(lr) - ∫ x l cos(lr) dx
Step 4: Simplify the remaining integral.
The remaining integral, ∫ x l cos(lr) dx, can be solved by substituting z = lr:
dz/dx = l
dx = dz/l
Now we can rewrite the integral in terms of z:
∫ z cos(z) dz
This integral can be solved using integration techniques such as integration by parts or integration tables.
Step 5: Solve the simplified integral.
Integrating z cos(z) with respect to z will give us -z sin(z) - cos(z) + C.
Step 6: Substitute back the original variable.
Substituting z = lr into the previous step, we have:
-lr sin(lr) - cos(lr) + C
Step 7: Combine the simplified integral with the previous terms.
Returning to the previous expression, we have:
x sin(lr) - (lr sin(lr) + cos(lr)) + C
Simplifying, we get the final answer:
-r cos(lnr) + sin(lr) + C
And that's how you integrate these expressions using the integration by parts method.