Integrate the following functions- -tan 22 x-3 d x

∫ tan^2 (2x 3)dx=∫ sec^2 (2x 3)dx ∫ 1dx (∵ tan^2 x=sec^2 x 1) Let 2x 3 =t ⇒ 2dx =dt ⇒ dx =1/2dt ∴∫ sec^2 (2x 3)dx ∫ dx =1/2∫ (sec^2 t)dt ∫ 1 dx =1/2tan t ...

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Standard XII

Mathematics

Integration by Substitution

Integrate the...

Question

Integrate the following functions.
$\int t a n^{2} (2 x - 3) d x .$

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Solution

$\int t a n^{2} (2 x - 3) d x = \int s e c^{2} (2 x - 3) d x - \int 1 d x (∵ t a n^{2} x = s e c^{2} x - 1)$
Let 2x-3 =t
$\Rightarrow 2 d x = d t \Rightarrow d x = \frac{1}{2} d t ∴ \int s e c^{2} (2 x - 3) d x - \int d x = \frac{1}{2} \int (s e c^{2} t) d t - \int 1 d x = \frac{1}{2} t a n t - x + C = \frac{1}{2} t a n (2 x - 3) - x + C = \frac{t a n (2 x - 3)}{2} - x + C$

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