Use the table of integral to evaluate: the integral of x^4 sinxdx. Please indicate the number of formula used.
How do you expect us to know what "table of integrals" you are using ?
It is up to you to find the pattern of the given question in your tables.
it's just the general table of integrals, most calc math book has it or online
To evaluate the integral of x^4 sin(x)dx, we can use integration by parts. Integration by parts is a technique that allows us to transform the integral of a product of functions into an integral that is easier to evaluate.
Step 1: Choose u and dv
In integration by parts, we choose one function to differentiate, denoted by u, and another function to integrate, denoted by dv. In this case, we choose u = x^4 and dv = sin(x)dx.
Step 2: Calculate du and v
We differentiate u to find du and integrate dv to find v.
du = 4x^3 dx (taking the derivative of u = x^4)
v = ∫sin(x)dx = -cos(x) (integrating dv = sin(x)dx)
Step 3: Apply the integration by parts formula
The integration by parts formula is ∫u dv = uv - ∫v du. Using the values from Steps 2, the formula becomes:
∫x^4 sin(x)dx = -x^4 cos(x) - ∫(-cos(x) * 4x^3)dx
Step 4: Simplify the equation
Simplifying the right side of the equation, we have:
∫x^4 sin(x)dx = -x^4 cos(x) - (-4)∫x^3 cos(x)dx
Step 5: Repeat the integration by parts
We can now apply integration by parts again, using the integral on the right side of the equation.
Choose u = x^3 and dv = cos(x)dx.
Step 6: Calculate du and v
Calculating du and v:
du = 3x^2 dx (taking the derivative of u = x^3)
v = ∫cos(x)dx = sin(x) (integrating dv = cos(x)dx)
Step 7: Apply the integration by parts formula
Using the integration by parts formula:
∫x^4 sin(x)dx = -x^4 cos(x) - (-4)(-x^3 sin(x) - 3∫x^2 sin(x)dx)
Step 8: Simplify the equation
Simplifying further, we have:
∫x^4 sin(x)dx = -x^4 cos(x) + 4x^3 sin(x) + 12∫x^2 sin(x)dx
Step 9: Repeat the integration by parts
We can continue applying integration by parts to evaluate the remaining integral.
Choose u = x^2 and dv = sin(x)dx.
Step 10: Calculate du and v
Calculating du and v:
du = 2x dx (taking the derivative of u = x^2)
v = ∫sin(x) dx = -cos(x) (integrating dv = sin(x)dx)
Step 11: Apply the integration by parts formula
Using the integration by parts formula:
∫x^4 sin(x)dx = -x^4 cos(x) + 4x^3 sin(x) + 12(-x^2 cos(x) - 2∫x cos(x)dx)
Step 12: Simplify the equation
Simplifying further, we have:
∫x^4 sin(x)dx = -x^4 cos(x) + 4x^3 sin(x) - 12x^2 cos(x) - 24∫x cos(x)dx
Step 13: Repeat the integration by parts
To evaluate the remaining integral, we apply integration by parts one last time.
Choose u = x and dv = cos(x)dx.
Step 14: Calculate du and v
Calculating du and v:
du = dx (taking the derivative of u = x)
v = ∫cos(x) dx = sin(x) (integrating dv = cos(x)dx)
Step 15: Apply the integration by parts formula
Using the integration by parts formula:
∫x^4 sin(x)dx = -x^4 cos(x) + 4x^3 sin(x) - 12x^2 cos(x) - 24(-x sin(x) + ∫sin(x)dx)
Step 16: Simplify the equation
Simplifying further, we have:
∫x^4 sin(x)dx = -x^4 cos(x) + 4x^3 sin(x) - 12x^2 cos(x) + 24x sin(x) - 24 (-cos(x))
Step 17: Combine like terms
Combining like terms, we get:
∫x^4 sin(x)dx = -x^4 cos(x) + 4x^3 sin(x) - 12x^2 cos(x) + 24x sin(x) + 24 cos(x)
So, the integral of x^4 sin(x)dx is equal to -x^4 cos(x) + 4x^3 sin(x) - 12x^2 cos(x) + 24x sin(x) + 24 cos(x).
Throughout the integration by parts process, we used the formula of integration by parts four times.