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QUESTION: 1

The integral of tan^{4}x is:

Solution:

Begin by rewriting ∫tan^{4}xdx as ∫tan^{2}xtan^{2}xdx.

Now we can apply the Pythagorean Identity, tan^{2}x+1=sec^{2}x, or tan^{2}x=sec^{2}x−1

∫tan^{2}x tan^{2}x dx = ∫(sec^{2}x−1)tan^{2}xdx

Distributing the tan^{2}x:

= ∫sec^{2}xtan^{2}x − tan^{2}xdx

Applying the sum rule:

= ∫sec^{2}xtan^{2}xdx − ∫tan^{2}xdx

We'll evaluate these integrals one by one.

**First Integral**

This one is solved using a

Let u = tanx

Applying the substitution,

Because u = tanx,

**Second Integral**

Since we don't really know what ∫tan^{2}xdx is by just looking at it, try applying the tan^{2}x = sec^{2}x−1

identity again:

∫tan^{2}xdx = ∫(sec^{2}x−1)dx

Using the sum rule, the integral boils down to:

∫sec^{2}xdx − ∫1dx

The first of these, ∫sec^{2}xdx, is just tanx + C.

The second one, the so-called "perfect integral", is simply x+C.

Putting it all together, we can say:

∫tan^{2}xdx = tanx + C − x + C

And because C+C is just another arbitrary constant, we can combine it into a general constant C:

∫tan^{2}xdx = tanx − x + C

Combining the two results, we have:

∫tan^{4}xdx=∫sec^{2}xtan^{2}xdx−∫tan^{2}xdx

=(tan^{3}x/3 + C) − (tanx − x + C)

=tan^{3}x/3 − tanx + x + C

Again, because C+C is a constant, we can join them into one C.

QUESTION: 2

Integrate

Solution:

∫(2+tan x)^{2}dx

= ∫(4 + tan^{2} x + 4tan x)dx

= ∫4 dx + ∫tan^{2} x dx + 4∫tan x dx

= 4x + ∫(sec^{2} x - 1)dx + 4(log|sec x|)

= 4x + tanx - x + 4(log|sec x|)

3x + tanx + 4(log|sec x|) + c

QUESTION: 3

Evaluate:

Solution:

∫sin^{2}(2x+1) dx

Put t = 2x+1

dt = 2dx

dx= dt/2

= 1/2∫sin^{2} t dt

=1/2∫(1-cos^{2}t)/2 dt

= 1/4∫(1-cos^{2}t) dt

= ¼[(t - (sin^{2}t)/2]dt

= t/4 - sin2t/8 + c

= (2x+1)/4 - ⅛(sin(4x+2)) + c

= x/2 - 1/8sin(4x+2) + ¼ + c

As ¼ is also a constant, so eq is = x/2 - 1/8sin(4x+2) + c

QUESTION: 4

The value of bb

Solution:

Let cos^{−1}(sinx)=θ

⇒ sinx=cosθ

⇒ sinx=sin(π/2−θ)

⇒ x = π/2−θ

⇒ θ = π/2−x

∴ ∫cos^{−1}(sinx)dx=∫(π/2−x)dx

= ∫π/2dx−∫xdx

= πx/2 - x^{2}/2 + c, where b is a constant of integration.

QUESTION: 5

Solution:

I = ∫cos2x/(sinx+cosx)^{2}dx

⇒I = ∫cos^{2}x−sin^{2}x(sinx+cosx)^{2}dx

⇒I = ∫[(cosx+sinx)(cosx−sinx)]/(sinx+cosx)^{2}dx

⇒I = ∫(cosx−sinx)/(sinx+cosx)dx

Let sinx+cosx = t

(cosx−sinx)dx = dt

Then, I = ∫dt/t

I = log|t|+c

I = log|sinx + cosx| + c

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