Question

1: Integration ( sinx+e^x)dx. 2: integration x^-3dx. 3:integration 2x^2dx please explain wit steps​

Answer 1

1)

Given:

A function

f(x)= sinx+eˣ

To Find:

Integration of f(x)

Solution:

Let the integration of f(x) be I

So,

$\longrightarrow\rm{I=\displaystyle\int{(sinx+e^{x})dx}}$

$\longrightarrow\rm\green{I=-cosx+e^{x}+c}$

where, c is any constant

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2)

Given:

A function

f(x)= x⁻³

To Find:

Integration of f(x)

Solution:

Let the integration of f(x) be I

So,

$\longrightarrow\rm{I=\displaystyle\int{(x^{-3})dx}}$

$\longrightarrow\rm{I=\dfrac{x^{-3+1}}{-3+1}+c}$

$\longrightarrow\rm{I=\dfrac{x^{-2}}{-2}+c}$

$\longrightarrow\rm\green{I=\dfrac{-1}{2x^{2}}+c}$

where, c is any constant

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3)

Given:

A function

f(x)= 2x²

To Find:

Integration of f(x)

Solution:

Let the integration of f(x) be I

So,

$\longrightarrow\rm{I=\displaystyle\int{(2x^{2})dx}}$

$\longrightarrow\rm{I=2\displaystyle\int{(x^{2})dx}}$

$\longrightarrow\rm{I=\dfrac{2x^{2+1}}{2+1}+c}$

$\longrightarrow\rm\green{I=\dfrac{2x^{3}}{3}+c}$

where, c is any constant

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Formulae to Remember

$\longrightarrow\rm{\displaystyle\int{(x^{n})dx}=\dfrac{x^{n+1}}{n+1}+c}$

$\longrightarrow\rm{\displaystyle\int{(c.f(x))dx}=c\displaystyle\int{(f(x))dx}}$

$\longrightarrow\rm{\displaystyle\int{(sinx)dx}=-cosx+c}$

$\longrightarrow\rm{\displaystyle\int{(e^{x})dx}=e^{x}+c}$

where, c is any constant

Answer 2

Answer -

1.

Given - $sinx + {e}^{x}$

Formula used -

$\longrightarrow$$\int f(x) + g(x) dx = \int f(x)dx + \int g(x)dx$

Solution -

Let $\longrightarrow$ $f(x) = sinx$

$\longrightarrow$$g(x) = {e}^{x}$

$\longrightarrow$$\int sinx = -cosx + c$

$\longrightarrow$$\int {e}^{x} dx = {e}^{x} + c$

$\longrightarrow$$\int f(x) + g(x) dx = - cosx + {e}^{x} + c$

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2.

Given - ${x}^{ - 3}$

Formula used -

$\longrightarrow$$\int {x}^{ n } dx = \frac{ {x}^{n + 1} }{n + 1}$

Solution -

$\longrightarrow$$\int {x}^{ - 3} = \frac{ {x}^{ - 2} }{ - 2} + c$

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3.

Given -$2 {x}^{2}$

Formula used -

$\longrightarrow$ $\int {x}^{ n } dx = \frac{ {x}^{n + 1} }{n + 1} + c$

Solution -

$\longrightarrow$$\int 2 {x}^{2} = 2 \times \frac{ {x}^{2 + 1} }{2 + 1}$

$\longrightarrow$= $2 \times \frac{ {x}^{3} }{3} + c$

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ADDITIONAL INFORMATION -

$\int dx = \int 1dx = x + c$

$\int sinx = -cosx + c$

$\int cosx = sinx + c$

$\int f(x) - g(x) dx=\int f(x)dx - \int g(x)dx$

Thanks