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Examples : Integration by Parts 1 Video Lecture | Mathematics (Maths) Class 12 - JEE

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FAQs on Examples : Integration by Parts 1 Video Lecture - Mathematics (Maths) Class 12 - JEE

1. What is integration by parts and how does it work?
Ans. Integration by parts is a technique used in calculus to evaluate the integral of a product of two functions. It is based on the product rule for differentiation. The formula for integration by parts is ∫ u dv = uv - ∫ v du, where u and v are differentiable functions. By selecting appropriate u and dv, we can simplify the integral by integrating v and differentiating u.
2. When should I use integration by parts?
Ans. Integration by parts is typically used when we have a product of two functions in the integrand. It is particularly helpful when one of the functions becomes simpler or easier to integrate after differentiating it, while the other function becomes simpler or easier to differentiate after integrating it. It is also useful when the integral involves trigonometric functions, logarithmic functions, or exponential functions.
3. Can you provide an example of integration by parts?
Ans. Certainly! Let's consider the integral ∫ x sin(x) dx. To solve this, we can choose u = x and dv = sin(x) dx. Then, du = dx and v = -cos(x). Applying the integration by parts formula, we have: ∫ x sin(x) dx = -x cos(x) - ∫ (-cos(x)) dx = -x cos(x) + sin(x) + C, where C is the constant of integration.
4. Are there any specific rules or guidelines to follow when applying integration by parts?
Ans. Yes, there are guidelines to follow when applying integration by parts: 1. Choose u and dv: Select u as a function that becomes simpler after differentiating, and dv as a function that becomes simpler after integrating. 2. Calculate du and v: Differentiate u to find du and integrate dv to find v. 3. Apply the integration by parts formula: Using the formula ∫ u dv = uv - ∫ v du, substitute the values of u, dv, du, and v into the formula. 4. Simplify and solve: Evaluate the integral on the right-hand side of the formula, simplifying as much as possible, and add the constant of integration if necessary.
5. Can integration by parts be used to solve definite integrals?
Ans. Yes, integration by parts can be used to solve definite integrals as well. After applying integration by parts to obtain a simplified expression, we can substitute the limits of integration and evaluate the definite integral. Just make sure to keep track of any additional terms that may arise from the integration by parts process, such as boundary terms, and include them in the final evaluation.
204 videos|288 docs|139 tests

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