Evaluate the following integral. | Basic Mathematics 2079

Basic Mathematics20795 marksNumerical: evaluating integralsBasic integration rules

Evaluate the following integral. $\int_{0}^{\frac{\pi}{4}} \sqrt{1 + \cos x}dx$ [5]

Evaluate the following integral. $$\int_{0}^{\frac{\pi}{4}} \sqrt{1 + \cos x}dx$$ [5]

Integral 1: $\displaystyle\int{0}^{\pi/4} \sqrt{1+\cos x}\,dx$ - Integral 2: $\displaystyle\int \frac{3x^2-7x+1}{3x}\,dx$ --- Step 1: Simplify using half-angle identity $1 + \cos x = 2\cos^2\frac{x}{2}$ So: $\sqrt{1+\cos x} = \sqrt{2}\left\cos\frac{x}{2}\right$ For $x \in [0, \pi/4]$ , we ha...

Integral 1: $\displaystyle\int{0}^{\pi/4} \sqrt{1+\cos x},dx$ - Integral 2: $\displaystyle\int \frac{3x^2-7x+1}{3x},dx$ --- Step 1: Simplify using half-angle identity $$1 + \cos x = 2\cos^2\frac{x}{2}$$ So: $$\sqrt{1+\cos x} = \sqrt{2}\left\cos\frac{x}{2}\right$$ For $x \in [0, \pi/4]$, we ha...