Integrate the following functions-2 cos x-3 sin x-6 cos x-4 sin x d x

∫2 cos x 3 sin x/6 cos x+4 sin xdx=∫2 cos x 3 sin x/2(3 cos x +2 sin x)dx Let 3 cos x +2 sin x=t ⇒ 3 sin x+2 cos x =dt/dx⇒ dx =dt/2 cos x 3 sin x ∴∫2 cos x ...

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Standard XII

Mathematics

Integration by Substitution

Integrate the...

Question

Integrate the following functions.
$\int \frac{2 c o s x - 3 s i n x}{6 c o s x + 4 s i n x} d x$

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Solution

$\int \frac{2 c o s x - 3 s i n x}{6 c o s x + 4 s i n x} d x = \int \frac{2 c o s x - 3 s i n x}{2 (3 c o s x + 2 s i n x)} d x$
Let 3 cos x +2 sin x=t
$\Rightarrow - 3 s i n x + 2 c o s x = \frac{d t}{d x} \Rightarrow d x = \frac{d t}{2 c o s x - 3 s i n x} ∴ \int \frac{2 c o s x - 3 s i n x}{2 (3 c o s x + 2 s i n x)} d x = \int \frac{2 c o s x - 3 s i n x}{2 (t)} \frac{d t}{2 c o s x - 3 s i n x} = \frac{1}{2} \int \frac{1}{t} d t = \frac{1}{2} l o g | t | + C = \frac{1}{2} l o g | 3 c o s x + 2 s i n x | + C$

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Integrate the following functions. ∫2cosx−3sinx6cosx+4sinxdx

∫2cosx−3sinx6cosx+4sinxdx=∫2cosx−3sinx2(3cosx+2sinx)dx Let 3 cos x +2 sin x=t ⇒−3sinx+2cosx=dtdx⇒dx=dt2cosx−3sinx∴∫2cosx−3sinx2(3cosx+2sinx)dx=∫2cosx−3sinx2(t)dt2cosx−3sinx=12∫1tdt=12log|t|+C=12log|3cosx+2sinx|+C

Integrate the following functions.
$\int \frac{2 c o s x - 3 s i n x}{6 c o s x + 4 s i n x} d x$