Indefinite Integration TOP 10 QUESTIONS

Question 1 : Solve `\int(x^2+3x-2)`dx

 = `\int(x^2+3x-2)`dx 

= `\[x^3/3 + 3/2x^2 -2x]`

=`\x^3/3 +3/2x^2 -2x + c`

Question 2 : Solve `\int 4^xe^(2x)` dx

= `\int4^xe^(2x)`dx

=`\int2^(2x)e^(2x)`dx

=`\int(2e)^(2x)`dx

=`[(2e)^(2x)]/{2+log4}` +c

Question 3 : Solve `\int{1+tan^2x}/{1-tan^2x}`

= `\int[1-tan^2x]/[sec^2x]`dx

= `\[int1/sec^2x - tan^2x/sec^2x ]`dx

= `\int(cos^2x-sin^2x)`dx

=`\intcos2x`dx

=`[sin2x]/2 +c`

Question 4 : Solve `\int 1/[1+cos6x]`dx

= `\int1/[2cos^2(3x)]`dx

=`\int1/2sec^2(3x)`dx

=`{tan(3x)}/6 +c`

Question 5: Solve `\intx^4/[1+x^10}` dx

= `\intx^4/[1+x^4]`dx

=`\intx^4/[1+(x^5)^2]`dx ..... 1

Let's use the subsitution method to solve the problem 

put x^5 = t, then 

Differente it

= `5x^4dx=dt`

=`x^4dx=dt/5`

Putting the value in equation 1, we have

=`\int1/[5(1+t)^2] = 1/5 tan^-1t + c`

=`1/5tan^-1x^5 +c`

Question 6 : Solve `\int [sin8x]/[1-cos^4(4x)]dx`

=`\int{sin8x}/\sqrt[1-(cos^2(4x))^2] dx` ......(1)

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