Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • explain solution RD Sharma class 12 Chapter 18 Indefinite Integrals exercise 18.30 question 15

explain solution RD Sharma class 12 Chapter 18 Indefinite Integrals exercise 18.30 question 15

Answers (1)

best_answer

Answer:

            \! \! \! \! \! \! \! \! \! \! \! \frac{a\left(a^{2}+b\right)+c}{(a-b)(a-c)} \log |x-a|+\frac{b^{2}(a+1)+c}{(b-a)(b-c)} \log |x-b|+\frac{c(a c+b+1)}{(c-a)(c-b)} \log |x-c|+k

Hint:

            To solve this integration, we use partial fraction method   

Given:

            \int \frac{a x^{2}+b x+c}{(x-a)(x-b)(x-c)} d x \\

Explanation:

\begin{aligned} &I=\int \frac{a x^{2}+b x+c}{(x-a)(x-b)(x-c)} d x \\ &\frac{a x^{2}+b x+c}{(x-a)(x-b)(x-c)}=\frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{x-c} \end{aligned}

\begin{aligned} &\frac{a x^{2}+b x+c}{(x-a)(x-b)(x-c)}=\frac{A(x-b)(x-c)+B(x-a)(x-c)+C(x-a)(x-b)}{(x-a)(x-b)(x-c)} \\ &a x^{2}+b x+c=A(x-b)(x-c)+B(x-a)(x-c)+C(x-a)(x-b) \end{aligned}

\begin{aligned} &\text { For } x=b \\ &a b^{2}+b^{2}+c=0+B(b-a)(b-c)+(0) \\ &\frac{(a+1) b^{2}+c}{(b-a)(b-c)}=B \\ &\text { For } x=a \end{aligned}

 

\begin{aligned} &a^{3}+b a+c=A(a-b)(a-c)+0+0 \\ &\frac{a^{3}+a b+c}{(a-b)(a-c)}=A \end{aligned}
\begin{aligned} &\text { For } x=c \\ &a c^{2}+b c+c=0+0+C(c-a)(c-b) \\ &\frac{c(a c+b+1)}{(c-a)(c-b)}=C \end{aligned}
\begin{aligned} &\frac{a x^{2}+b x+c}{(x-a)(x-b)(x-c)}=\frac{a\left(a^{2}+b\right)+c}{(a-b)(a-c)} \cdot \frac{1}{x-a}+\frac{b^{2}(a+1)+c}{(b-a)(b-c)} \cdot \frac{1}{x-b}+\frac{c(a c+b+1)}{(c-a)(b-c)} \cdot \frac{1}{x-c} \\ &I=\frac{a\left(a^{2}+b\right)+c}{(a-b)(a-c)} \int \frac{1}{x-a} d x+\frac{b^{2}(a+1)+c}{(b-a)(b-c)} \int \frac{1}{x-b} d x+\frac{c(a c+b+1)}{(c-a)(c-b)} \int \frac{1}{x-c} d x \\ &I=\frac{a\left(a^{2}+b\right)+c}{(a-b)(a-c)} \log |x-a|+\frac{b^{2}(a+1)+c}{(b-a)(b-c)} \log |x-b|+\frac{c(a c+b+1)}{(c-a)(c-b)} \log |x-c|+k \end{aligned}

 

 

Posted by

infoexpert27

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

CBSE OSM controversy deepens as students allege fresh errors, seek grace marks
anam-khan

CBSE to issue revised Class-12 physical marksheets through regional offices
anam-khan

NEET aspirant Aditya Mishra becomes CBSE Science topper after re-evaluation

Latest Question