Question

integrate with the limits √ln2 to √ln3 x*sinx^2/sinx^2+sin(ln6-x^2) dx​

Answer 1

EXPLANATION.

$\sf \implies \displaystyle \int_{\sqrt{ln2} }^{\sqrt{ln3} } \dfrac{x sin(x^{2} )}{sin(x^{2} ) + sin(ln 6 - x^{2} )} dx$

As we know that,

We can use substitution method in this equation, we get.

Let, we assume that.

⇒ x² = t.

Differentiate w.r.t x, we get.

⇒ 2x dx = dt.

⇒ x dx = dt/2.

As we know that,

In definite integration, if we apply substitution method then limit will also change, we get.

First, we put the lower limit, we get.

⇒ (√㏑2)² = t.

⇒ ㏑2 = t. = [new lower limit].

⇒ (√㏑3)² = t.

⇒ ㏑3 = t. = [new upper limit].

Put the values in the equation, we get.

$\sf \implies \displaystyle \int_{{ln2} }^{{ln3} } \dfrac{sin(t)}{sin(t) + sin(ln 6 - t)} \dfrac{dt}{2}$

$\sf \implies I = \dfrac{1}{2} \displaystyle \int_{{ln2} }^{{ln3} } \dfrac{sin(t)}{sin(t) + sin(ln 6 - t)} dt. - - - - - (1).$

As we know that,

Formula of :

$\sf \implies \int\limits^b_a {f(x)} \, dx = \int\limits^b_a {f(a + b - x)} \, dx$

Proof :

Replace : x = a + b - x.

$\sf \implies \int\limits^b_a {f(x)} \, dx$

⇒ x = a + b - x.

⇒ dx = - dt.

$\sf \implies \int\limits^a_b {f(a + b - x)} \,- dt$

a = a + b - t.

$\sf \implies \int\limits^b_a {f(a+ b - t) } \, dt$

Replace t = x in the equation, we get.

$\sf \implies \int\limits^b_a {f(a + b - x)} \, dx$

Hence Proved.

Replace,

⇒ t = ㏑3 + ㏑2 - t.

⇒ t = ㏑6 - t.

$\sf \implies I = \dfrac{1}{2} \displaystyle \int_{{ln2} }^{{ln3} } \dfrac{sin(ln 6 - t)}{sin(ln 6 - t) + sin(t)} dt. - - - - - (2).$

Adding equation (1) and (2), we get.

$\sf \implies 2I = \dfrac{1}{2} \displaystyle \int_{{ln2} }^{{ln3} } \dfrac{sin(t) + sin(ln 6 - t)}{sin(t )+ sin(ln 6 - t)} dt.$

$\sf \implies 2I = \dfrac{1}{2} \displaystyle \int_{{ln2} }^{{ln3} } dt.$

Putt the upper and lower limit in the equation, we get.

$\sf \implies 2I = \dfrac{1}{2} \displaystyle \bigg[t \bigg]_{ln 2}^{ln 3}$

$\sf \implies 2I = \dfrac{1}{2} \displaystyle \bigg[ ln 3 - ln 2 \bigg]$

$\sf \implies I = \dfrac{1}{4} ln \dfrac{3}{2}$

                                                                                                                       

MORE INFORMATION.

Properties of definite integration.

$\sf (1) = \int\limits^b_a {f(x)} \, dx = \int\limits^b_a {f(t)} \, dt$

$\sf (2) = \int\limits^b_a {f(x)} \, dx = - \int\limits^a_b {f(x)} \, dx$

$\sf (3) = \int\limits^b_a {f(x)} \, dx = \int\limits^c_a {f(x)} \, dx + \int\limits^b_c {f(x)} \, dx \ \ where \ \ a < c < b.$

$\sf (4) = \int\limits^a_0 {f(x)} \, dx = \int\limits^a_0 {f(a - x)} \, dx$

$\sf (5) = \int\limits^b_a {f(x)} \, dx = \int\limits^b_a {f(a + b - x)} \, dx$

Answer 2

✯ SOLUTION ✯

Given in attachment.

߷ ABOUT INTEGRATION ߷

The integration is the process of finding the antiderivative of a function. It is a similar way to add the slices to make it whole. The integration is the inverse process of differentiation.The integration is used to find the volume, area and the central values of many things.

HISTORY

Niels Henrik Abel seems to have been the first to consider in a general way the question as to what differential equations can be integrated in a finite form by the aid of ordinary functions, an investigation extended by Liouville. Cauchy early undertook the general theory of determining definite integrals, and the subject has been prominent during the 19th century. Frullani integrals, David Bierens de Haan's work on the theory and his elaborate tables, Lejeune Dirichlet's lectures embodied in Meyer's treatise, and numerous memoirs of Legendre, Poisson, Plana, Raabe, Sohncke, Schlömilch, Elliott, Leudesdorf and Kronecker are among the noteworthy contributions.

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