USE INTEGRATION BY PARTS TO FIND EACH INTEGRAL
ƪxe^2x dx
To find the integral of xe^2x, we can use integration by parts. The integration by parts formula is given by:
∫u * dv = uv - ∫v * du
Let's assign u and dv as follows:
u = x
dv = e^2x dx
Now, we can find du and v by differentiating and integrating the assigned values:
du = dx
v = ∫dv = ∫e^2x dx
To find the integral of e^2x, we can use the formula for integrating exponential functions:
∫e^kx dx = (1/k) * e^kx + C
In this case, k = 2. Applying this formula, we have:
v = (1/2) * e^(2x)
Now, we can substitute these values into the integration by parts formula:
∫xe^2x dx = uv - ∫v * du
= x * (1/2) * e^(2x) - ∫(1/2)e^(2x) dx
Simplifying this, we get:
∫xe^2x dx = (1/2) * x * e^(2x) - (1/2) * ∫e^(2x) dx
Now, we need to find the integral of e^(2x):
∫e^(2x) dx = (1/2) * e^(2x) + C
Replacing this value back into our equation, we get:
∫xe^2x dx = (1/2) * x * e^(2x) - (1/2) * (1/2) * e^(2x) + C
= (1/2) * [x - (1/2)] * e^(2x) + C
Therefore, the integral of xe^2x is equal to (1/2) * [x - (1/2)] * e^(2x) + C.
To solve the integral ƪxe^2x dx using integration by parts, we need to apply the integration by parts formula.
The integration by parts formula states:
∫ u dv = uv - ∫ v du
Step 1: Identify u and dv
Let u = x and dv = e^2x dx
Step 2: Compute du and v
To find du, we differentiate u with respect to x:
du/dx = 1
Therefore, du = dx
To find v, we integrate dv:
v = ∫ e^2x dx
Step 3: Calculate v
To integrate v = ∫ e^2x dx, we can recognize that the derivative of e^2x is e^2x itself multiplied by a constant, which means that the integral of e^2x is e^2x divided by that constant.
Using this fact, we can divide both sides of v = ∫ e^2x dx by 2 to find:
v = (1/2)e^2x
Step 4: Apply the integration by parts formula
Using the integration by parts formula mentioned earlier, we have:
∫ xe^2x dx = uv - ∫ v du
Substituting the values we found earlier:
∫ xe^2x dx = x * (1/2)e^2x - ∫ (1/2)e^2x dx
Simplifying, we get:
∫ xe^2x dx = (1/2)xe^2x - (1/2) ∫ e^2x dx
Step 5: Calculate the remaining integral
The remaining integral, ∫ e^2x dx, can be calculated using the method of substitution or integration by parts again. However, we can recognize that it is a simple integral of e^2x, which we solved for when calculating v.
∫ e^2x dx = (1/2)e^2x + C (where C is the constant of integration)
Step 6: Final answer
Substituting this value back into the equation, we have:
∫ xe^2x dx = (1/2)xe^2x - (1/2) * [(1/2)e^2x + C]
Simplifying further, we get the final answer:
∫ xe^2x dx = (1/2)xe^2x - (1/4)e^2x - C