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# Exercise 7.6

Integrate the following equation

Let

Integrating by parts, we obtain

Again integrating by parts, we obtain

Integrate the function

Let

Taking x2 as first function and ex as second function and integrating by parts, we obtain

Again integrating by parts, we obtain

Integrate the function

x sin x

Let I =

Taking x as first function and sin x as second function and integrating by parts, we obtain

Integrate the function

Let I =

Taking x as first function and sin 3x as second function and integrating by parts, we obtain

Integrate the function

x logx

Let

Taking log x as first function and x as second function and integrating by parts, we obtain

Integrate the function

x log 2x

Let

Taking log 2x as first function and x as second function and integrating by parts, we obtain

Integrate the function

xlog x

Let

Taking log x as first function and x2 as second function and integrating by parts, we obtain

Integrate the function

Let

Taking as first function and x as second function and integrating by parts, we obtain

Integrate the function

Let

Taking  as first function and x as second function and integrating by parts, we obtain

Integrate the function

Let

Taking cos−1 x as first function and x as second function and integrating by parts, we obtain

Integrate the function

Let

Taking as first function and 1 as second function and integrating by parts, we obtain

Integrate the function

Let

Taking  as first function and  as second function and integrating by parts, we obtain

Integrate the function

Let

Taking x as first function and sec2x as second function and integrating by parts, we obtain

Integrate the function

Let

Taking  as first function and 1 as second function and integrating by parts, we obtain

Integrate the function

Taking  as first function and x as second function and integrating by parts, we obtain

Integrate the function

Let

Let I = I1 + I2 … (1)

Where, and

Taking log x as first function and xas second function and integrating by parts, we obtain

Taking log x as first function and 1 as second function and integrating by parts, we obtain

Using equations (2) and (3) in (1), we obtain

Integrate the function

Treat the function currently as A * B (A= $e^{x}$ , B = (sin x + cos x). I am sure, you must be aware how to integrate integral (A*B).  Now further for integral B : treat it as integral of C+D , where C = sinx , D = cosx.

First apply basic formula for integral A*B, and dont hurry to replace with actual values. Then solve integral of B as integral of C+D.  Once you have values in A, C, D, then only replace.

Let

⇒

∴

It is known that,

Integrate the function

Let

Let  ⇒

It is known that,

Integrate the function

Let ⇒

It is known that,

From equation (1), we obtain

Integrate the function

Also, let  ⇒

It is known that,

Integrate the function

Let  ⇒

It is known that,

Integrate the function

Let ⇒

= 2θ

⇒

Integrating by parts, we obtain

equals

Let

Also, let  ⇒

Hence, the correct answer is A.

equals

Let

Also, let  ⇒

Hence, the correct answer is A.

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