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NCERT Solutions Class 12 Maths (integrals) Miscellaneous Exercise

NCERT Solutions Class 12 Maths (integrals) Miscellaneous Exercise

NCERT Solutions Class 12 Maths from class 12th Students will get the answers of Chapter-7(integrals) Miscellaneous Exercise 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.
Solutions Class 12 Maths (integrals) Miscellaneous Exercise

Q1. Integrate the function 

Answer. 1xx3=1x(1x2)=1x(1x)(1+x)  Let 1x(1x)(1+x)=Ax+B(1x)+C1+x1=A(1x2)+Bx(1+x)+Cx(1x)1=AAx2+Bx+Bx2+CxCx2 Equating the coefficients of x2 , x, and constant term, we obtain A+BC=0B+C=0A=1 On solving these equations, we obtain A=1,B=12, and C=12 From equation (1), we obtain 1x(1x)(1+x)=1x+12(1x)12(1+x) 1x(1x)(1+x)=1x+12(1x)12(1+x)1x(1x)(1+x)dx=1xdx+1211xdx1211+xdx 

Q2. Integrate the function 

Answer. 1x+a+x+b=1x+a+x+b×x+ax+bx+ax+b=x+ax+b(x+a)(x+b)=(x+ax+b)ab 

Q3. Integrate the function 

Answer. 1xaxx2 Let x=atdx=at2dt 1xaxx2dx=1ataat(at)2(at2dt)=1at11t1t2dt =1a1t2tt2t2dt=1a1t1dt=1a[2t1]+C=1a[2ax1]+C 

Q4. Integrate the function 

Answer. 1x2(x4+1)34 Multiplying and dividing by x3 , we obtain x3x2x3(x4+1)34=x3(x4+1)34x2x3 =(x4+1)34x5(x4)34=1x5(x4+1x4)34=1x5(1+1x4)34  Let 1x4=t4x5dx=dt1x5dx=dt41x2(x4+1)34dx=1x5(1+1x4)34dx =14(1+t)34dt=14[(1+t)1414]+C 

Q5. Integrate the function 

Answer. 1x12+x13=1x13(1+x16) Let x=t6dx=6t5dt 1x12+x13dx=1x13(1+x16)dx=6t5t2(1+t)dt=6t3(1+t)dt  On dividing, we obtain 1x12+x13dx=6{(t2t+1)11+t}dt 

Q6. Integrate the function 

Answer.  Let 5x(x+1)(x2+9)=A(x+1)+Bx+C(x2+9).(1)5x=A(x2+9)+(Bx+C)(x+1)5x=Ax2+9A+Bx2+Bx+Cx+C  Equating the coefficients of x2,x, and constant term, we obtain A+B=0B+C=59A+C=0  On solving these equations, we obtain A=12,B=12, and C=92 From equation (1), we obtain