Question

Solve this integral...​

Answer 1

Question :

$\displaystyle \sf \red{\int \dfrac{d \theta}{(sin\ \theta - 2 cos\ \theta)(2\ sin\ \theta + cos\ \theta )}}$

Solution :

$\\ \to \displaystyle \sf \int \dfrac{d \theta}{ \bigg( \frac{cos\ \theta}{cos\ \theta} \bigg) (sin\ \theta - 2 cos\ \theta) \bigg( \frac{cos\ \theta}{cos\ \theta} \bigg) (2\ sin\ \theta + cos\ \theta )}$

$\\ \to \displaystyle \sf \int \dfrac{d \theta }{(cos^2 \theta )(tan\ \theta - 2)(2\ tan\ \theta + 1)}$

$\\ \bullet\ \; \pink{ \sf sec^2 \theta = \dfrac{1}{ cos^2 \theta }}$

$\\ \to \displaystyle \sf \int \dfrac{sec^2 \theta\ d \theta }{(tan\ \theta - 2)(2\ tan\ \theta + 1)}$

Let's use substitution ,

⇒ u = tan θ

⇒ du = sec²θ dθ

$\\ \to \displaystyle \sf \int \dfrac{du}{(u-2)(2u+1)}$

Use partial faction to evaluate , ( Find attachment )

$\to \displaystyle \sf \dfrac{1}{5} \int \dfrac{du}{u-2} - \dfrac{2}{5} \int \dfrac{du}{2u+1}$

$\to \displaystyle \sf \dfrac{1}{5} . \ln (u-2) - \dfrac{2}{5} . \dfrac{ \ln (2u+1)}{2}+c$

$\to \sf \dfrac{1}{5} \ln (u - 2) - \dfrac{1}{5} \ln (2\ u + 1)+c$

$\to \sf \dfrac{1}{5} \bigg( \ln (u -2) - \ln (2u + 1) \bigg)+c$

$\bullet\ \; \sf \pink{ \ln A - \ln B = \dfrac{ \ln A}{ \ln B}}$

$\to \sf \dfrac{1}{5}\ ln\bigg[ \dfrac{u-2}{ 2u+1} \bigg]+c$

• u = tan θ

$\to \sf \green{\dfrac{1}{5}\ \ln\bigg[ \dfrac{ tan\ \theta -2}{ 2\ tan\ \theta+1} \bigg]+c }$

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$\displaystyle \sf \int \dfrac{d \theta}{(sin\ \theta - 2 cos\ \theta)(2\ sin\ \theta + cos\ \theta )} = \dfrac{1}{5}\ \ln\bigg[ \dfrac{ tan\ \theta -2}{ 2\ tan\ \theta+1} \bigg]+c$

Answer 2

Answer:

Temples were constructed as places of worship and are meant to demonstrate the power, wealth and devotion of the patron. The temples were miniature models of the world ruled by the king and his allies. For example, the Rajarajeshvara temple was built by King Rajarajadeva for the worship of his god, Rajarajeshvaram. Generally, the names of the rulers and the gods were very similar. The king took the god’s name because it was auspicious and he wanted to appear like a God. Through the rituals of worship in the temple, one god (Rajarajadeva) honored another (Rajarajeshvaram).