Calculus problem - Evaluate the indefinite integral: ∫0 to π/4 (tan^6(x)sec^2(x)dx)?

Question

This is an internet homework, and the answer does have +c in it. I have no idea how to do this since its asking for indefinite, but then it gives me from 0 to pi/4.... any ideas?

Answer 1

I get the answer 1/7.

I split the integral up like this:

tan^5x tan x sec^2x dx. We know that d/dx tan x is sec^2 x. By integration by susbstitution we get

u^5 * u du because u = tan x and d/du = sec^2 x.

then we integrate u^6 to get

1/7 u^7 = 1/7 tan^7x | from u = 0 and u = (pi/4)

then we evaluate to get 1/7

Answer 2

Been a at the same time as; combine with the aid of areas enable u=x^2-a million then du= (2x)dx resolve int (u^4)du from 0 to a million supplies (u^5)/5 then substititute back the u: that's [(x^2-a million)^5]/5 evaluated from 0 to a million: whilst x=a million the int is 0, whilst x=0 the int is -a million/5 -a million/5 - 0 = -a million/5 answer

Answer 3

this is a definite integral, so like you said, no constant c.

the easiest way to do this problem is substitution

u = tanx so du = sec^2(x)dx

change your limits of integration
if x = 0, then u = tan(0) = 0
if x = pi/4, then u = tan(pi/4) = 1

now the new integral is

integral from 0 to 1 of u^6du

= (1/7)u^7 with limits of int from 0 to 1

I hope that this helps

Answer 4

Evaluate the integral over the interval [0, π/4].

∫(tan^6 x)(sec²x)dx
Let
u = tan x
du = sec²x dx
____

= ∫u^6 du = (1/7)u^7 = (1/7)(tan x) | [Eval from 0 to π/4]

= (1/7)[tan(π/4) - tan(0)] = (1/7)(1 - 0) = 1/7