Q2) Find the integral value for the following functions...

Q2) Find the integral value for the following functions (choose five only): 1. ( int frac{3x}{sqrt{1+3x^2}} dx ) 2. ( int (sec(4x) - 1)^2 dx ) 3. ( int frac{dx}{1+(5x+1)^2} ) 4. ( int frac{dx}{x^2sqrt{x^2-1}} ) 5. ( int frac{e^{cos^{-1}x}}{sqrt{1-x^2}} dx ) 6. ( int frac{5}{x} sin(ln x) dx )

Transcript

00:01 Hi, now we are going to integrate the following.

00:04 The first question is integral over 3x divided by square root of 1 plus 3x squared dx.

00:15 Now we take u is equal to 3x squared plus 1.

00:20 Then d u is equal to 6x into dx.

00:25 Then the given integral is equal to 1 by 2 integral over 3x 1 by root u into d u and it can be written as 1 by 2 into 2 root u plus c and it will be equal to root u plus c then it can be written as root of 3 x squared plus 1 plus c therefore the answer for the first question is root of 3 x squared plus 1 plus c.

01:11 Now the second question is integral over 2x minus 1 the whole squared into dx.

01:20 Now let us take u equal to 4x, then du is equal to 4dx, then the given integral can be written as integral over 2cun u minus 1 the whole square du.

01:38 It will be equal to tan u minus 2 ln of tan u plus second u plus u plus u plus c integral over second 4x minus 1 the whole squared inter d x is equal to tan 4x minus 2 ln of tan 4x tan 4x minus 2 ln of tan 4 4 4x plus 2x 2x plus 4x plus 4x plus c and the third question is integral over d x divide by 1 plus 5x plus 1 the whole square in let us take u equal to 5x plus 1 then d u is equal to 5 d x x then the given integral can be written as 1 1 by 5 integral over 1 by u squared plus 1 into du after integration it will be equal to 1 by 5 10 inverse of u plus c therefore integral over d x divide by 1 plus 5 x divide by 1 plus 5 x plus 1 the whole square is equal to 1 by 5 5 tan inverse of 5x plus 1 plus c.

03:22 And next, the fourth question is integral over dx divide by x squared into root of x squared minus 1.

03:32 Now we take x equal to second u...

Question

Q2) Find the integral value for the following functions (choose five only): 1. ( int frac{3x}{sqrt{1+3x^2}} dx ) 2. ( int (sec(4x) - 1)^2 dx ) 3. ( int frac{dx}{1+(5x+1)^2} ) 4. ( int frac{dx}{x^2sqrt{x^2-1}} ) 5. ( int frac{e^{cos^{-1}x}}{sqrt{1-x^2}} dx ) 6. ( int frac{5}{x} sin(ln x) dx )

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