Class 11 Exam > Class 11 Questions > Integrate - rootx(x^3/2-7x^5/2 9x^7/2)dx?
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Integrate - rootx(x^3/2-7x^5/2 9x^7/2)dx?
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Integrate - rootx(x^3/2-7x^5/2 9x^7/2)dx?
Integration of √x(x3/2-7x5/2+9x7/2)dx
To solve this integration problem, we can use the power rule of integration and substitution method.
Step 1: Simplify the expression
We can simplify the expression inside the square root by factoring out x3/2:
√x(x3/2-7x5/2+9x7/2)dx = √x(x3/2(1-7x+9x2))dx
Step 2: Substitute a new variable
Let's substitute u = 1-7x+9x2 and solve for x:
u = 1-7x+9x2
9x2-7x+(1-u) = 0
Using the quadratic formula, we get:
x = (7 ± √(49-36(1-u)))/18
x = (7 ± 5√u)/18
dx = (5/18) √u du
Step 3: Substitute u and simplify
Now, we can substitute u and dx in the integral expression:
∫ √x(x3/2-7x5/2+9x7/2)dx = ∫ √x(x3/2(1-7x+9x2))dx
= ∫ √x(x3/2u)dx
= ∫ x3/4u1/2(5/18) √u du
= (5/18) ∫ u1/2x3/4du
= (5/18) ∫ (1-7x+9x2)1/2x3/4du
This integral expression cannot be simplified further using elementary functions. Therefore, the final answer is:
∫ √x(x3/2-7x5/2+9x7/2)dx = (5/18) ∫ (1-7x+9x2)1/2x3/4du
To solve this integration problem, we can use the power rule of integration and substitution method.
Step 1: Simplify the expression
We can simplify the expression inside the square root by factoring out x3/2:
√x(x3/2-7x5/2+9x7/2)dx = √x(x3/2(1-7x+9x2))dx
Step 2: Substitute a new variable
Let's substitute u = 1-7x+9x2 and solve for x:
u = 1-7x+9x2
9x2-7x+(1-u) = 0
Using the quadratic formula, we get:
x = (7 ± √(49-36(1-u)))/18
x = (7 ± 5√u)/18
dx = (5/18) √u du
Step 3: Substitute u and simplify
Now, we can substitute u and dx in the integral expression:
∫ √x(x3/2-7x5/2+9x7/2)dx = ∫ √x(x3/2(1-7x+9x2))dx
= ∫ √x(x3/2u)dx
= ∫ x3/4u1/2(5/18) √u du
= (5/18) ∫ u1/2x3/4du
= (5/18) ∫ (1-7x+9x2)1/2x3/4du
This integral expression cannot be simplified further using elementary functions. Therefore, the final answer is:
∫ √x(x3/2-7x5/2+9x7/2)dx = (5/18) ∫ (1-7x+9x2)1/2x3/4du
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Integrate - rootx(x^3/2-7x^5/2 9x^7/2)dx?
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Integrate - rootx(x^3/2-7x^5/2 9x^7/2)dx? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Integrate - rootx(x^3/2-7x^5/2 9x^7/2)dx? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Integrate - rootx(x^3/2-7x^5/2 9x^7/2)dx?.
Integrate - rootx(x^3/2-7x^5/2 9x^7/2)dx? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Integrate - rootx(x^3/2-7x^5/2 9x^7/2)dx? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Integrate - rootx(x^3/2-7x^5/2 9x^7/2)dx?.
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