Blog

Parin Sir > Mathematical Proofs > Indefinite Integration > Pattern - 24 of Indefinite Integration > Pattern - 24 of Indefinite Integration

Pattern - 24 of Indefinite Integration

No Comments

⋆ Pattern – 24

\int\frac{p\cos{x}+q\sin{x}+r}{a\cos{x}+b\sin{x}+n}\ dx,

\int\frac{p\cos{x}+q\sin{x}}{a\cos{x}+b\sin{x}}\ dx,\ \int\frac{p\cos{x}}{a\cos{x}+b\sin{x}}\ dx

\int\frac{q\sin{x}}{a\cos{x}+b\sin{x}}dx,

then express the number as:

Nr = A( D_r )+ B \frac{d}{dx} ( D_r ) + c

1) \ I=\ \int\frac{5\sin{x+6}}{\sin{x+2\cos{x}+3}}\ dx

Let,

(5sin x + 6) = A( sin x + 2 cos x + 3 ) + B( cos x – 2 sin x ) + c

∴  5 sin x + 6 = (A – 2B) sin x + (2A + B)cos x + (3A + c)

∴  A – 2B = 5 —(1)

2A + B = 0 ——(2)

3A + c = 6 ——(3)

Here,

A – 2B = 5

4A + 2B = 0

5A = 5

∴  A = 1

∴  from 2

2 + B = 0

∴  B = -2

∴  from 3

3 + c = 6

∴  c = 3

\therefore\ \ I=\ \int\frac{1\times\ \left(\sin{x}+2\cos{x}+3\right)-2\ \left(\cos{x}-2\sin{x}\right)+3}{\sin{x}+2\cos{x}+3}\ dx

\therefore\ \ I=\ \int1dx-2\int\frac{\cos{x}-2\sin{x}}{\sin{x}+2\cos{x}+3}\ dx+\int\frac{3}{\sin{x}+2\cos{x}+3}\ dx

=\ \int1dx-2-2\ \int\frac{\cos{x}-2\sin{x}}{\sin{x}+2\cos{x}+3}\ dx+3\ I_1

Where,\ I_1=\ \int\frac{1}{\sin{x}+2\cos{x}+3}\ dx

Let,\tan{\frac{x}{2}}=t

\therefore\ \ dx=\ \frac{2dt}{1+t^2},\sin{x}=\ \frac{2t}{1+t^2},\cos{x}=\ \frac{1-t^2}{1+t^2}

Here, \ I_1=\ \int\frac{1}{\sin{x}+2\cos{x}+3}\ dx

Let, \tan{\frac{x}{2}}=t

\therefore\ \ I_1=\ \int\frac{1}{\frac{2t}{1+t^2}+\ \frac{2\left(1-t^2\right)}{1+t^2}+3}\times\frac{2dt}{1+t^2}

=\ \int\frac{1+t^2}{2t+2-2t^2+3+3t^2}\times\frac{2dt}{1+t^2}

= 2\ \int\frac{1}{t^2+5t+2}\ dt

=2\ \int\frac{1}{t^2+5t+\ \frac{25}{4}-\frac{25}{4}+2}\ dt

=2\ \int\frac{1}{\left(t+\frac{5}{2}\right)^2-\frac{17}{4}}\ dt

=2\times\ \frac{1}{2\times\frac{\sqrt{17}}{2}}\times\log{\left|\frac{t+\frac{5}{2}-\ \frac{\sqrt{17}}{2}}{t+\frac{5}{2}+\ \frac{\sqrt{17}}{2}}\right|}

= \ \frac{2}{\sqrt{17}}\log{\left|\frac{2t+5-\ \sqrt{17}}{2t+5+\ \sqrt{17}}\right|}

=\ \int1dx-2\int\frac{\cos{x}-2\sin{x}}{\sin{x}+2\cos{x}+3}\ dx+3\ I_1---\left(2\right)

Now,

= \ \frac{2}{\sqrt{17}}\log{\left|\frac{2t+5-\ \sqrt{17}}{2t+5+\sqrt{17}}\right|}

= \ \frac{2}{\sqrt{17}}\log{\left|\frac{2\tan{\frac{x}{2}}+5-\ \sqrt{17}}{2\tan{\frac{x}{2}}+5+\ \sqrt{17}}\right|}

∴   from (2)

= x-2\log{\left|\sin{x}+2\cos{x}+3\right|}+\ \frac{6}{\sqrt{17}}\log{\left|\frac{2\tan{\frac{x}{2}}+5-\sqrt{17}}{2\tan{\frac{x}{2}}+5+\ \sqrt{17}}\right|}+c

 

Click here to see next pattern

Share
maths proof pattern sabiti

Leave a Reply