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integration of xlogx^2 dx
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integration of xlogx^2 dx
first we apply the log identity logx^2=2logxso 2xlogxdxby using u*v method 2 (logx*xdx)2 [{logx* integrated xdx}-{integration ofxdx * differentiation of logx dy/dx}integration dx]2 [logx*x^2/2-(x^2/2*1/x)dx]2logx*x^2/2-2*x^2/2*1/2+cso, x^2logx-x^2/2+c Ans.
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integration of xlogx^2 dx
Integration of xlogx^2 dx
To integrate the function xlogx^2 dx, we can use integration by parts. This method involves breaking down the integral into two parts and applying a specific formula.
Integration by parts formula:
∫ u dv = uv - ∫ v du
Let's apply this formula to the given function.
Since du = (1/x) dx, we can find v by integrating dv.
∫ x dx = (1/2)x^2 + C
Thus, v = (1/2)x^2 + C
Using the integration by parts formula:
∫ xlogx^2 dx = (1/2)x^2logx^2 - ∫ (1/2)x^2(1/x) dx
Simplifying the integral on the right side:
∫ (1/2)x dx = (1/4)x^2 + C
Now, let's substitute the values back into the original equation:
∫ xlogx^2 dx = (1/2)x^2logx^2 - (1/4)x^2 + C
And there we have it, the integration of xlogx^2 dx.
Summary:
- The integration of xlogx^2 dx can be solved using integration by parts.
- We choose u and dv such that their product makes the integral easier to solve.
- We then apply the integration by parts formula: ∫ u dv = uv - ∫ v du.
- After integrating dv, we find v, and after differentiating u, we find du.
- We substitute these values back into the formula and simplify the integral.
- Finally, we substitute the values back into the original equation to obtain the solution.
Integration by Parts Steps:
1. Choose u and dv.
2. Find du by differentiating u.
3. Find v by integrating dv.
4. Substitute the values into the integration by parts formula: ∫ u dv = uv - ∫ v du.
5. Simplify the integral on the right side.
6. Substitute the values back into the original equation to obtain the solution.
Final Result:
∫ xlogx^2 dx = (1/2)x^2logx^2 - (1/4)x^2 + C
To integrate the function xlogx^2 dx, we can use integration by parts. This method involves breaking down the integral into two parts and applying a specific formula.
Integration by parts formula:
∫ u dv = uv - ∫ v du
Let's apply this formula to the given function.
Since du = (1/x) dx, we can find v by integrating dv.
∫ x dx = (1/2)x^2 + C
Thus, v = (1/2)x^2 + C
Using the integration by parts formula:
∫ xlogx^2 dx = (1/2)x^2logx^2 - ∫ (1/2)x^2(1/x) dx
Simplifying the integral on the right side:
∫ (1/2)x dx = (1/4)x^2 + C
Now, let's substitute the values back into the original equation:
∫ xlogx^2 dx = (1/2)x^2logx^2 - (1/4)x^2 + C
And there we have it, the integration of xlogx^2 dx.
Summary:
- The integration of xlogx^2 dx can be solved using integration by parts.
- We choose u and dv such that their product makes the integral easier to solve.
- We then apply the integration by parts formula: ∫ u dv = uv - ∫ v du.
- After integrating dv, we find v, and after differentiating u, we find du.
- We substitute these values back into the formula and simplify the integral.
- Finally, we substitute the values back into the original equation to obtain the solution.
Integration by Parts Steps:
1. Choose u and dv.
2. Find du by differentiating u.
3. Find v by integrating dv.
4. Substitute the values into the integration by parts formula: ∫ u dv = uv - ∫ v du.
5. Simplify the integral on the right side.
6. Substitute the values back into the original equation to obtain the solution.
Final Result:
∫ xlogx^2 dx = (1/2)x^2logx^2 - (1/4)x^2 + C
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integration of xlogx^2 dx for CA Foundation 2025 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about integration of xlogx^2 dx covers all topics & solutions for CA Foundation 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for integration of xlogx^2 dx.
integration of xlogx^2 dx for CA Foundation 2025 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about integration of xlogx^2 dx covers all topics & solutions for CA Foundation 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for integration of xlogx^2 dx.
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