Question

Determine ∫tan8x sec4 x dx

please give the right answer

Answer 1

Answer:

Given: ∫tan8x sec4 x dx

Let I = ∫tan8x sec4 x dx — (1)

Now, split sec4x = (sec2x) (sec2x)

Now, substitute in (1)

I = ∫tan8x (sec2x) (sec2x) dx

= ∫tan8x (tan2 x +1) (sec2x) dx

It can be written as:

= ∫tan10x sec2 x dx + ∫tan8x sec2 x dx

Now, integrate the terms with respect to x, we get:

I =( tan11 x /11) + ( tan9 x /9) + C

Hence, ∫tan8x sec4 x dx = ( tan11 x /11) + ( tan9 x /9) + C

Question 3:

Write the anti-derivative of the following function: 3x2+4x3

Solution:

Given: 3x2+4x3

The antiderivative of the given function is written as:

∫3x2+4x3 dx =  3(x3/3) + 4(x4/4)

= x3 + x4

Thus, the antiderivative of 3x2+4x3 = x3 + x4