Evaluate the following integrals: 3 ^2x dx - Vidyadip

Question

Evaluate the following integrals:
$\int3\sqrt{\cos^2\text{x}}\sin\text{x }\text{dx}$

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Answer

Let I $=\int3\sqrt{\cos^2\text{x}}\sin\text{x }\text{dx}\ ....(1)$
Let $\cos\text{x}=\text{t}$ then,
$\text{d}(\cos\text{x})=\text{dt}$
$\Rightarrow-\sin\text{x}\text{ dx}=\text{dt}$
$\Rightarrow\text{dx}=-\frac{\text{dt}}{\sin\text{x}}$
Putting $\cos\text{x}=\text{t}$ and $\text{dx}=-\frac{\text{dt}}{\sin\text{x}}$ in equation (1), we get
$\text{I}=\int3\sqrt{\text{t}^2}\sin\text{x}\times\frac{-\text{dt}}{\sin\text{x}}$
$=-\int\text{t}^\frac{2}{3}\sin\text{x}\frac{\text{dt}}{\sin\text{x}}$
$=-\int\text{t}^\frac{2}{3}\text{dt}$
$=-\frac{3}{5}\times\text{t}^\frac{5}{3}+\text{C}$
$=-\frac{3}{5}(\cos\text{x})^\frac{5}{3}+\text{C}$
$\therefore\text{I}=-\frac{3}{5}(\cos\text{x})^\frac{5}{3}+\text{C}$

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