Question

Questi
Define ſtan x. dx =
secu
cotx
Secx.fans
log(secx)​

Answer 1

Answer:

Here the given integral ∫f(x) dx can be transformed into another form by changing the independent variable ‘x’ to ‘t’ by substituting x=g(t).

Consider I = ∫f(x) dx

Put x=g(t) so that dx/dt = g’(t)

We can write dx = g’(t)dt

Hence I = ∫f(x) dx = ∫f[g(t)] g’(t)dt

This change of variable formula is one of the important tools available to us in the name of integration by substitution. Usually, we make a substitution for a function whose derivative also occurs in the integrand.

We can have a glance at few examples.

i) indefiniteintegrals2-ii

= - log|t| + C

= -log|cosx| + C

= log|cosx|-1 + C

= log|1/cosx| + C

= log|secx| + C

Hence ∫tanx dx = log|secx| + C

Similarly we can find the integral of cotx also

∫cotx dx = log|sinx| + C

ii) indefiniteintegrals3-ii

Here we know the derivative of logx is 1/x, so put logx = t

log x = t

1/x dx = dt

indefiniteintegrals3-ii = ∫t2 dt

= t3/3 + C

= indefiniteintegrals4-ii+ C

Using substitution technique, we can find the following standard integrals.

i) ∫tanx dx = log |secx| + C

ii) ∫cotx dx = log |sinx| + C

iii) ∫secx dx = log |secx + tanx| + C

iv) ∫cosecx dx = log | cosecx - cotx| + C