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  • Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 25

Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 25

Answers (1)

Answer:\frac{1}{a^{2}-b^{2}}\log_{e}\left | a^{2}\sin^{2}x+b^{2}\cos ^{2}x \right |+c

Hints: You must know about the integral rule of trigonometric functions

Given: \int \frac{\sin2x}{a^{2}\sin ^{2}x+b^{2}\cos^{2}x}dx

Solution: \int \frac{\sin2x}{a^{2}\sin ^{2}x+b^{2}\cos^{2}x}dx

Let

a^{2} \sin ^{2} x+b^{2} \cos ^{2} x=t \\ a^{2} 2 \sin x \cdot \cos x d x+b^{2} 2 \cos x(-\sin x)=d t \\                        \\\quad \quad \quad \quad \quad \begin{bmatrix} \frac{d}{d x} \sin x=\cos x \\\\ \frac{d}{d x} \cos x=-\sin x \end{bmatrix}\&\\\left[\frac{d}{d x} x^{n}=n x^{n-1}\right]

\begin{aligned} &\sin 2 x \cdot d x\left(a^{2}-b^{2}\right)=d t \\ &\sin 2 x d x=\frac{d t}{\left(a^{2}-b^{2}\right)} \\ &{[2 \sin x \cos x=\sin 2 x]} \\ &\int \frac{d t}{\left(a^{2}-b^{2}\right) t} \\ &\frac{1}{a^{2}-b^{2}} \int \frac{d t}{t} \\ &\int \frac{1}{x} d x=\log |x|+c \\ &=\frac{1}{a^{2}-b^{2}} \log _{e}|t|+c \\ &=\frac{1}{a^{2}-b^{2}} \log _{e}\left|a^{2} \sin ^{2} x+b^{2} \cos ^{2} x\right|+c \end{aligned}

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