Answer :

Ideas required to solve the problems:

* __Integration by substitution__: A change in the variable of integration often reduces an integral to one of the fundamental integration. If derivative of a function is present in an integration or if chances of its presence after few modification is possible then we apply integration by substitution method.

* Knowledge of integration of fundamental functions like sin, cos ,polynomial, log etc and formula for some special functions.

Let, I =

To solve such integrals involving trigonometric terms in numerator and denominators. We use the basic substitution method and to apply this simply we follow the undermentioned procedure-

If I has the form

Then substitute numerator as -

Where A, B and C are constants

We have, I =

As I matches with the form described above, So we will take the steps as described.

∴

⇒ {

⇒

Comparing both sides we have:

3B+ C = 6

2B + A = 5

B - 2A = 0

On solving for A ,B and C we have:

A = 1, B = 2 and C = 0

Thus I can be expressed as:

I =

I =

∴ Let I_{1} = and I_{2} =

⇒ I = I_{1} + I_{2} ….equation 1

I_{1} =

Let, 2 cos x + sin x + 3 = u

⇒ (-2sin x + cos x)dx = du

So, I_{1} reduces to:

I_{1} =

∴ I_{1} = …..equation 2

As, I_{2} =

⇒ I_{2} = 2 …..equation 3

From equation 1, 2 and 3 we have:

I = +

∴ I =

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