Integrate the following functions-sin a x-b cos a x-b d x

Let I =∫ sin (ax+b)cos (ax+b)dx Let sub (ax+b)=t On differentiating w.r.t.x, we get a cos (ax+b)=dt/dx ⇒ dx=dt/a cos (ax+b) ∴ I =∫ t cos (ax +b)dt/a cos (ax+b)= ...

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Byju's Answer

Standard XII

Mathematics

Substitution Method to Remove Indeterminate Form

Integrate the...

Question

Integrate the following functions.
$\int$ sin (ax+b)cos (ax+b)dx.

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Solution

Let $I = \int s i n (a x + b) c o s (a x + b) d x$
Let sub (ax+b)=t
On differentiating w.r.t.x, we get a cos $(a x + b) = \frac{d t}{d x}$
$\Rightarrow d x = \frac{d t}{a c o s (a x + b)} ∴ I = \int t c o s (a x + b) \frac{d t}{a c o s (a x + b)} = \frac{1}{a} \int t d t = \frac{1}{a} . \frac{t^{2}}{2} + c = \frac{[s i n (a x + b)]^{2}}{2 a} + C$

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Integrate the following functions. ∫ sin (ax+b)cos (ax+b)dx.

Let I=∫sin(ax+b)cos(ax+b)dx Let sub (ax+b)=t On differentiating w.r.t.x, we get a cos (ax+b)=dtdx ⇒dx=dtacos(ax+b)∴I=∫tcos(ax+b)dtacos(ax+b)=1a∫tdt=1a.t22+c=[sin(ax+b)]22a+C

Integrate the following functions.
$\int$ sin (ax+b)cos (ax+b)dx.