Question

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Write the formulas of integration. Give some examples too. ( With conditions, if any )

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Thank You !!​

Answer 1

Integration:

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Integration is the reverse process of differentiation.

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$\large{\boxed{\red{\sf{\displaystyle\int y\:dx}}}}$

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$\sf{\bold{Here-}}$ $\begin{cases}\text{ \int is elongated S which means summation of the quantity}\\ \text{y is the function}\\ \text{x is the operator}\end{cases}$

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$\rule{200}2$

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Formulae:

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★ $\large{\sf{\displaystyle\int x^{n} \:dx = \dfrac{x^{n+1}}{n+1} + C\:(Where\:n \neq -1)}}$

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★ $\large{\sf{\displaystyle\int x^{-1} dx = log_{e} x}}$

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★ $\large{\sf{\displaystyle\int e^{x} dx = e^{x} + C}}$

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★ $\large{\sf{\displaystyle\int dx = x + C}}$

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★ $\large{\sf{\displaystyle\int Cos\:x.dx = Sin\:x + C}}$

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★ $\large{\sf{\displaystyle\int Sin\:x.dx = -Cos\:x + C}}$

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★ $\large{\sf{\displaystyle\int Sec^{2} x.dx = tan\:x + C}}$

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★ $\large{\sf{\displaystyle\int Cosec^{2} x.dx = -Cot\:x + C}}$

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★ $\large{\sf{\displaystyle\int Sec\:x\:Tan\:x.dx = Sec\:x + C}}$

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★ $\large{\rm{\displaystyle\int Cosec\:x\:Cot\:x.dx = -Cosec\:x + C}}$

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★ $\large{\sf{\displaystyle\int K.u.dx= K \int u.dx}}$

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★ $\large{\sf{\displaystyle\int (u \pm v) dx = \displaystyle \int u.dx \pm \displaystyle \int v.dx}}$

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★ $\large{\sf{\displaystyle \int uv.dx = u \displaystyle \int v.dx - \displaystyle \int \left[ \dfrac{du}{dx} \displaystyle \int v.dx \right] dx}}$

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★ $\large{\sf{\displaystyle \int k.dx = kx + C}}$

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Definite Integration:

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Example:

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$\large{\sf{\displaystyle \int\limits_{2}^{3} x^{2} dx}}$

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$\implies$ $\sf{\left[ \dfrac{(3)^3}{3} - \dfrac{(2)^3}{3}\right]}$

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$\implies$ $\sf{\left[ 9 - \dfrac{8}{3}\right]}$

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$\implies$ $\blue{\sf{\dfrac{19}{3}}}$

Answer 2

Answer:

In this section, we use some basic integration formulas studied previously to solve some key applied problems. It is important to note that these formulas are presented in terms of indefinite integrals. Although definite and indefinite integrals are closely related, there are some key differences to keep in mind. A definite integral is either a number (when the limits of integration are constants) or a single function (when one or both of the limits of integration are variables). An indefinite integral represents a family of functions, all of which differ by a constant. As you become more familiar with integration, you will get a feel for when to use definite integrals and when to use indefinite integrals. You will naturally select the correct approach for a given problem without thinking too much about it. However, until these concepts are cemented in your mind, think carefully about whether you need a definite integral or an indefinite integral and make sure you are using the proper notation based on your choice.

Basic Integration Formulas

Recall the integration formulas given in the table in Antiderivatives and the rule on properties of definite integrals. Let’s look at a few examples of how to apply these rules.