Evaluate the integral using th eigven substitution f(3/x^2) sin^2 (3/x) dx, u =-

Question

Evaluate the integral using th eigven substitution

f(3/x^2) sin^2 (3/x) dx, u =-3/x

Find the derivative

sqrt(x)

d/dx f 18t^5 dt

1

X

y= f sqrt(8x+7) dt

0

X

y=   f dt/3t+6

0

Find the area enclosed by the given curves.

2) y = 2x - x2, y = 2x - 4

Use the substitution formula to evaluate the integral.

pi/2

f   cot x csc3 x dx

pi/6

3pi/4

f ( 4+ cot O) csc^2 O dO

pi/4

) Suppose that

4 0

f f(x) dx=13 Find   f f(x) dx, if f is odd

0 -4

Find the area enclosed by the given curves.

7) y = - 4sin x, y = sin 2x, 0 <= x <= pi

Explanation / Answer

du/dx = 5
so du = 5dx
we know 15 = 3 times 5.
so you can put the 5 multiplied by dx and leave the 3 out the front. This makes it easier to sub in du.
hence
%u222B 15(5x-6)^-3 dx = %u222B3 (5x - 6)^-3 5dx = 3%u222B(u)^-3 du (by subbing in u=.. and 5dx =... )
then by using %u222B x^n dx = (n+1)x^(n+1) which is on the table of standard integrals.
3%u222B(u)^-3 du = 3[(u^-2)/-2] + c
sub back in for u
= (3(5x - 6)^-2)/-2 + c
= 3/(-2(5x - 6)^2) + c

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