Question

Evaluate the value of the following integral: $\int{\dfrac{\sin x+\cos x}{\sqrt{1+2\sin 2x}}dx}$

Answer 1

Hint: To evaluate the value of following integral square the expression $\sin x+\cos x$ and simplify it using the identity ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$. Further, simplify the expression using the trigonometric identities ${{\cos }^{2}}x+{{\sin }^{2}}x=1$ and $\sin 2x=2\sin x\cos x$. Simplify the integral and then use the fact that $\int{{{x}^{n}}dx}=\dfrac{{{x}^{n+1}}}{n+1}$.

Step-by-step answer:
We have to evaluate the value of the integral $\int{\dfrac{\sin x+\cos x}{\sqrt{1+2\sin 2x}}dx}$.
To do so, we will first simplify the expression $\sqrt{1+2\sin 2x}$.
We will firstly square the expression $\sin x+\cos x$. We know the algebraic identity ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$.
Substituting $a=\cos x,b=\sin x$ in the above equation, we have ${{\left( \sin x+\cos x \right)}^{2}}={{\sin }^{2}}x+{{\cos }^{2}}x+2\sin x\cos x$.
We know the trigonometric identities ${{\cos }^{2}}x+{{\sin }^{2}}x=1$ and $\sin 2x=2\sin x\cos x$.
Thus, we can rewrite the above equation as ${{\left( \sin x+\cos x \right)}^{2}}={{\sin }^{2}}x+{{\cos }^{2}}x+2\sin x\cos x=1+\sin 2x.....\left( 1 \right)$.
Substituting equation (1) in the integral $\int{\dfrac{\sin x+\cos x}{\sqrt{1+2\sin 2x}}dx}$, we have $\int{\dfrac{\sin x+\cos x}{\sqrt{1+2\sin 2x}}dx}=\int{\dfrac{\sin x+\cos x}{\sqrt{{{\left( \sin x+\cos x \right)}^{2}}}}dx}$.
Simplifying the above equation, we have $\int{\dfrac{\sin x+\cos x}{\sqrt{1+2\sin 2x}}dx}=\int{\dfrac{\sin x+\cos x}{\sqrt{{{\left( \sin x+\cos x \right)}^{2}}}}dx}=\int{\dfrac{\sin x+\cos x}{\sin x+\cos x}dx}=\int{1dx}$.
We know that the integral of ${{x}^{n}}$ is $\int{{{x}^{n}}dx}=\dfrac{{{x}^{n+1}}}{n+1}$.
Substituting $n=0$ in the above expression, we have $\int{{{x}^{0}}dx}=\int{1dx}=\dfrac{x}{1}=x$.
Hence, the value of the integral $\int{\dfrac{\sin x+\cos x}{\sqrt{1+2\sin 2x}}dx}$ is $x$.

Note: We observe that this is an indefinite integral. An indefinite integral is a function that takes the antiderivative of another function. It represents a family of functions whose derivatives are the function given in the integral. It’s necessary to use the trigonometric identities to evaluate the value of the given integral. Otherwise; we won’t be able to solve the question.