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  • Please Solve RD Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.15 Question 2 Maths Textbook Solution.

Please Solve RD Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.15 Question 2 Maths Textbook Solution.

Answers (1)

Answer:

            \frac{1}{3} \tan ^{-1}\left(\frac{x-5}{3}\right)+C

Hint:

            To solve this problem use special integration formula

Given:

            \int \frac{1}{x^{2}-10 x+34} d x

Solution:

Let    I=\int \frac{1}{x^{2}-10 x+34} d x

            \begin{aligned} &=\int \frac{1}{x^{2}-2 \cdot x \cdot 5+5^{2}-5^{2}+34} d x \\\\ &=\int \frac{1}{(x-5)^{2}-25+34} d x=\int \frac{1}{(x-5)^{2}+9} d x \\\\ &=\int \frac{1}{(x-5)^{2}+3^{2}} d x \end{aligned}

Put      x-5=t \Rightarrow d x=d t

Then    I=\int \frac{1}{t^{2}+3^{2}} d t

            =\frac{1}{3} \tan ^{-1}\left(\frac{t}{3}\right)+C                                \quad\left[\because \int \frac{1}{x^{2}+a^{2}} d x=\frac{1}{a} \tan ^{-1}\left(\frac{x}{a}\right)+C\right]

            =\frac{1}{3} \tan ^{-1}\left(\frac{x-5}{3}\right)+C                          \quad[\because t=x-5]

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