**Mathematics**

#
Evaluate the following definite integrals as limit of sums.

$$\displaystyle\int^b_axdx$$.

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**Asked in: **Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

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**A.**$$3$$ -
**B.**$$0$$ -
**C.**$$4$$ -
**D.**$$6$$

**Asked in: **Mathematics - Integrals

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View Answer$$\int {\frac{e^x(x-1)}{(x+1)^3}dx}$$

**Asked in: **Mathematics - Integrals

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View Answer**Asked in: **Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

View AnswerIn calculating a number of integrals we had to use the method of integration by parts several times in succession. The result could be obtained more rapidly and in a more concise form by using the so-called generalized formula for integration by parts.

$$\int u(x)\, v(x)dx\, =\, u(x)\, v_{1}(x)\, -\, u^{}(x)v_{2}(x)\, +\, u^{}(x)\, v_{3}(x)\, -\, .\, +\, (-1)^{n\, -\, 1}u^{n\, -\, 1}(x)v_{n}(x)\, -\, (-1)^{n\, -\, 1}$$ $$\int\, u^{n}(x)v_{n}(x)\, dx$$ where $$v_{1}(x)\, =\, \int v(x)dx,\, v_{2}(x)\, =\, \int v_{1}(x)\, dx\, ..\, v_{n}(x)\, =\, \int v_{n\, -\, 1}(x) dx$$

Of course, we assume that all derivatives and integrals appearing in this formula exist. The use of the generalized formula for integration by parts is especially useful when calculating $$\int P_{n}(x)\, Q(x)\, dx$$, where $$P_{n}(x)$$, is polynomial of degree n and the factor Q(x) is such that it can be integrated successively n + 1 times.

**Asked in: **Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

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