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NCERT Solutions Class 12 Maths Chapter-7 (Integrals)Exercise 7.5

NCERT Solutions Class 12 Maths Chapter-7 (Integrals)Exercise 7.5

NCERT Solutions Class 12 Maths from class 12th Students will get the answers of Chapter-7 (Integrals)Exercise 7.5 This chapter will help you to learn the basics and you should expect at least one question in your exam from this chapter.
We have given the answers of all the questions of NCERT Board Mathematics Textbook in very easy language, which will be very easy for the students to understand and remember so that you can pass with good marks in your examination.
NCERT Solutions Class 12 maths Chapter-7 (Integrals)Exercise 7.5

Exercise 7.5

Q1. Integrate the rational function : 

Answer. x Let x(x+1)(x+2)=A(x+1)+B(x+2)x=A(x+2)+B(x+1) Equating the coefficients of x and constant term, we obtain  A+B=12A+B=0 On solving, we obtain A=1 and B=2 x(x+1)(x+2)=1(x+1)+2(x+2)x(x+1)(x+2)dx=1(x+1)+2(x+2)dx 

Q2. Integrate the rational function : 

Answer. 1(x+3)(x3)=A(x+3)+B(x3)1=A(x3)+B(x+3)  Equating the coefficients of x and constant term, we obtain A+B=03A+3B=1  On solving, we obtain A=16 and B=161(x+3)(x3)=16(x+3)+16(x3) 

Q3. Integrate the rational function : 

Answer. 3x1(x1)(x2)(x3)=A(x1)+B(x2)+C(x3)3x1=A(x2)(x3)+B(x1)(x3)+C(x1)(x2) Substituting x=1,2, and 3 respectively in equation (1), we obtain  A=1,B=5, and C=43x1(x1)(x2)(x3)=1(x1)5(x2)+4(x3) 

Q4. Integrate the rational function : 

Answer. xLetx(x1)(x2)(x3)=A(x1)+B(x2)+C(x3)x=A(x2)(x3)+B(x1)(x3)+C(x1)(x2)...(1) Substituting x=1,2, and 3 respectively in equation (1), we obtain  A=12,B=2, and C=32x(x1)(x2)(x3)=12(x1)2(x2)+32(x3) 

Q5. Integrate the rational function : 

Answer. 2xx2+3x+2=A(x+1)+B(x+2)2x=A(x+2)+B(x+1) Substituting x=1 and 2 in equation (1), we obtain A=2 and B=4 

Q6. Integrate the rational function : 

Answer.  It can be seen that the given integrand is not a proper fraction.  Therefore, on dividing (1x2) by x(12x), we obtain 1x2x(12x)=12+12(2xx(12x)) 2xx(12x)=Ax+B(12x)(2x)=A(12x)+Bx Substituting x=0 and 12 in equation (1), we obtain  A=2 and B=32xx(12x)=2x+312x Substituting in equation (1), we obtain  1x2x(12x)=12+12{2x+3(12x)}1x2x(12x)dx={12+12(2x+312x)}dx 

Q7. Integrate the rational function : 

Answer. x(x2+1)(x1)=Ax+B(x2+1)+C(x1)x=(Ax+B)(x1)+C(x2+1)x=Ax2Ax+BxB+Cx2+C  Equating the coefficients of x2,x, and constant term, we obtain A+C=0A+B=1B+C=0  On solving these equations, we obtain A=12,B=12, and C=12 From equation (1), we obtain  x(x2+1)(x1)=(12x+12)x2+1+12(x1)x(x2+1)(x1)=12xx2+1dx+121x2+1dx+121x1dx =142xx2+1dx+12tan1x+12log|x1|+C Consider 2xx2+1dx, let (x2+1)=t2xdx=dt2xx2+1dx=dtt=log|t|=log|x2+1|