- #1
intelli
- 20
- 0
Homework Statement
integral limit 1 to 5
integral of sqrt x * lnx dx
a = 1
b= 5
Homework Equations
The Attempt at a Solution
2
x (-1 + 2 Log[x])
------------------
8
11.99604193 but its not right
rock.freak667 said:Not sure how you reached that answer but did you use
u=lnx and dv=x1/2 dx ?
intelli said:so is this right
du = 1/x dx
v = 3/2x^3/2
and plug into the parts formula?
rock.freak667 said:v=2/3 x3/2
check that back.
then yes put that into the formula.
intelli said:yes so i get this is this right after integrating
2/3 x^3/2 ln x (between limits 1 to 5 ) - 4/9 x^3/2 (between limits 1 to 5) and
i get 9.602
Integration by parts is a mathematical technique used to find the integral of a product of two functions. It is based on the product rule of differentiation and can be used to simplify integration problems that would otherwise be difficult or impossible to solve.
Integration by parts should be used when the integral of a function cannot be easily evaluated by other methods, such as substitution or trigonometric identities. It is particularly useful for integrals involving products of polynomials, exponential functions, or trigonometric functions.
To use integration by parts, you must first identify the functions in the integral that can be differentiated and integrated. Then, you use the product rule to rewrite the integral as a product of two functions. Finally, you apply the integration by parts formula and solve for the integral.
The integration by parts formula is ∫u dv = uv - ∫v du, where u and v are the two functions in the integral and du and dv represent their respective derivatives.
Yes, integration by parts can be used for definite integrals. In this case, the integration by parts formula becomes ∫a to b u dv = [uv]a to b - ∫a to b v du, where a and b are the limits of integration. The resulting integral can then be evaluated using the fundamental theorem of calculus.