Question

Question 3: e^2x

Class 12 - Math - Integrals

Answer 1

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Answer 2

EXPLANATION.

⇒ ∫e²ˣ dx.

As we know that,

In this type of integration we can use substitution reaction, we get.

Let we assume that,

⇒ 2x = t.

Differentiate w.r.t x, we get.

⇒ 2 dx = dt.

⇒ dx = dt/2.

Put the value in equation, we get.

$\sf \implies \int (e^{t}) \dfrac{dt}{2}$

As we know that,

We can take constant term outside from integration, we get.

$\sf \implies \dfrac{1}{2} \displaystyle \int (e^{t}) dt$

As we know that,

Formula of :

⇒ ∫eˣ dx = eˣ + c.

Using this formula in equation, we get.

$\sf \implies \dfrac{1}{2} \bigg[e^{t} \ + c \bigg].$

Substitute the value of t = 2x in equation, we get.

$\sf \implies \dfrac{e^{2x} }{2} \ + c.$

$\sf \implies \int e^{2x} dx = \dfrac{e^{2x} }{2} \ + c.$

                                                                                                                           

MORE INFORMATION.

Integration by parts.

(1) = If u and v are two functions of x then,

∫(u v )dx = u (∫v dx) - ∫[(du/dx). ∫v dx)]dx.

From the first letter of the word.

I = Inverse trigonometric functions.

L = Logarithmic functions.

A = Algebraic functions.

T = Trigonometric functions.

E = Exponential functions.

This is known as = ILATE.

First arrange the functions in the order according to the letters of this word and then integrate by parts.

(2) = If the integrals is of the form,

⇒ ∫eˣ [f(x) + f'(x)]dx = eˣ f(x) + c.

(3) = If the integrals is on the form,

⇒ ∫[x f'(x) + f(x)]dx = x f(x) + c.