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  1. A personal development coach in Palm Beach helps individuals unlock their full potential by providing guidance, motivation, and strategies for self-improvement. Whether you’re looking to enhance your career, build confidence, improve relationships, or set and achieve meaningful goals, a coach can prRead more

    A personal development coach in Palm Beach helps individuals unlock their full potential by providing guidance, motivation, and strategies for self-improvement. Whether you’re looking to enhance your career, build confidence, improve relationships, or set and achieve meaningful goals, a coach can provide the clarity and structure needed for success. For more information, please visit: https://www.vanessa-gray.com/

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Priyansh Srivastava
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∫(∏r=0 to m (1/(x+r)))dx , find the value of this integral

  1. This answer was edited.

    To evaluate the integral: \[\int \prod_{r=0}^{m} \frac{1}{x + r} \, dx\] we can proceed with the following steps: Step 1: Express the Product as a SumThe integrand is a product of terms of the form \(\frac{1}{x + r}\). To simplify the integration, we can use partial fraction decomposition. Assume thRead more

    To evaluate the integral:

    \[
    \int \prod_{r=0}^{m} \frac{1}{x + r} \, dx
    \]

    we can proceed with the following steps:

    Step 1: Express the Product as a Sum
    The integrand is a product of terms of the form \(\frac{1}{x + r}\). To simplify the integration, we can use partial fraction decomposition. Assume that:

    \[
    \prod_{r=0}^{m} \frac{1}{x + r} = \sum_{r=0}^{m} \frac{A_r}{x + r}
    \]

    where \(A_r\) are constants to be determined.

    Step 2: Determine the Constants \(A_r\)
    Multiply both sides by \(\prod_{r=0}^{m} (x + r)\):

    \[
    1 = \sum_{r=0}^{m} A_r \prod_{\substack{k=0 \\ k \neq r}}^{m} (x + k)
    \]

    To find \(A_r\), set \(x = -r\). This eliminates all terms in the sum except the one corresponding to \(A_r\):

    \[
    1 = A_r \prod_{\substack{k=0 \\ k \neq r}}^{m} (-r + k)
    \]

    Simplify the product:

    \[
    A_r = \frac{1}{\prod_{\substack{k=0 \\ k \neq r}}^{m} (k – r)}
    \]

    This can be written as:

    \[
    A_r = \frac{(-1)^r}{r! (m – r)!}
    \]

    Step 3: Integrate Term by Term
    Now, the integral becomes:

    \[
    \int \sum_{r=0}^{m} \frac{A_r}{x + r} \, dx = \sum_{r=0}^{m} A_r \int \frac{1}{x + r} \, dx
    \]

    The integral of \(\frac{1}{x + r}\) is \(\ln|x + r|\), so:

    \[
    \sum_{r=0}^{m} A_r \ln|x + r| + C
    \]

    Substitute \(A_r\):

    \[
    \sum_{r=0}^{m} \frac{(-1)^r}{r! (m – r)!} \ln|x + r| + C
    \]

    Step 4: Simplify the Expression
    The sum can be written in terms of binomial coefficients:

    \[
    \sum_{r=0}^{m} \frac{(-1)^r}{r! (m – r)!} \ln|x + r| = \frac{1}{m!} \sum_{r=0}^{m} (-1)^r \binom{m}{r} \ln|x + r|
    \]

    Thus, the final result is:

    \[
    \boxed{\frac{1}{m!} \sum_{r=0}^{m} (-1)^r \binom{m}{r} \ln|x + r| + C}
    \]

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Pankaj Gupta
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In which year was the Indian Independence Act passed?

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  1. The Indian Independence Act was passed on 18th July 1947 by the British Parliament. This act paved the way for the partition of India and the creation of two independent dominions, India and Pakistan, effective from 15th August 1947. It marked the end of British rule in India and granted both nationRead more

    The Indian Independence Act was passed on 18th July 1947 by the British Parliament. This act paved the way for the partition of India and the creation of two independent dominions, India and Pakistan, effective from 15th August 1947. It marked the end of British rule in India and granted both nations the power to govern themselves.

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Pankaj Gupta
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The Doctrine of Lapse was introduced by which British Governor-General?

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  1. The Doctrine of Lapse was introduced by Lord Dalhousie, who served as the Governor-General of India from 1848 to 1856. This policy allowed the British East India Company to annex Indian princely states if a ruler died without a natural male heir, disregarding the traditional practice of adopting heiRead more

    The Doctrine of Lapse was introduced by Lord Dalhousie, who served as the Governor-General of India from 1848 to 1856. This policy allowed the British East India Company to annex Indian princely states if a ruler died without a natural male heir, disregarding the traditional practice of adopting heirs. Under this doctrine, several states, including Satara (1848), Jaitpur (1849), Sambalpur (1850), Udaipur (1852), Jhansi (1853), and Nagpur (1854), were annexed by the British. The policy was widely resented and became one of the causes of the Revolt of 1857.

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Pankaj Gupta
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Which ancient Indian text is considered the first comprehensive work on Ayurveda?

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  1. The Charaka Samhita is considered the first comprehensive work on Ayurveda. It is an ancient Indian text attributed to Acharya Charaka, who is regarded as one of the principal contributors to Ayurvedic medicine. The text primarily focuses on internal medicine (Kaya Chikitsa) and provides detailed knRead more

    The Charaka Samhita is considered the first comprehensive work on Ayurveda. It is an ancient Indian text attributed to Acharya Charaka, who is regarded as one of the principal contributors to Ayurvedic medicine. The text primarily focuses on internal medicine (Kaya Chikitsa) and provides detailed knowledge about diagnosis, treatment, pharmacology, and preventive healthcare. It is believed to have been compiled around the 2nd century BCE and is one of the foundational texts of Ayurveda, alongside the Sushruta Samhita (which focuses on surgery) and the Ashtanga Hridaya.

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Pankaj Gupta
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Who among the following was the first woman to win a Nobel Prize?

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  1. The first woman to win a Nobel Prize was Marie Curie. She won the Nobel Prize in Physics in 1903, which she shared with her husband Pierre Curie and physicist Henri Becquerel for their work on radioactivity. Later, she won the Nobel Prize in Chemistry in 1911 for her discovery of radium and poloniumRead more

    The first woman to win a Nobel Prize was Marie Curie. She won the Nobel Prize in Physics in 1903, which she shared with her husband Pierre Curie and physicist Henri Becquerel for their work on radioactivity. Later, she won the Nobel Prize in Chemistry in 1911 for her discovery of radium and polonium, making her the first person to win two Nobel Prizes in different scientific fields.

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