Find -x2-sin 2 x 2 x-1-x2 x d x

Let I = ∫(x^2 +sin^2 x)sec^2 x/1+x^2 dx.=∫(1+x^2+2sin^2 x 1)sec^2 x/1+x^2dx ⇒ I =∫(1+x^2 cos^2 x)sec^2 x/1+x^2 dx =∫(1+x^2)sec^2 x 1/1+x^2dx ⇒ I =∫ [ sec^2 x ...

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Byju's Answer

Standard XII

Mathematics

Sign of Trigonometric Ratios in Different Quadrants

Find ∫x2+sin ...

Question

Find $\int \frac{(x^{2} + s i n^{2} x) s e c^{2} x}{1 + x^{2}} d x$ .

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Solution

Let $I = \int \frac{(x^{2} + s i n^{2} x) s e c^{2} x}{1 + x^{2}} d x . = \int \frac{(1 + x^{2} + 2 s i n^{2} x - 1) s e c^{2} x}{1 + x^{2}} d x$
$\Rightarrow I = \int \frac{(1 + x^{2} - c o s^{2} x) s e c^{2} x}{1 + x^{2}} d x = \int \frac{(1 + x^{2}) s e c^{2} x - 1}{1 + x^{2}} d x$
$\Rightarrow I = \int [s e c^{2} x - \frac{1}{1 + x^{2}}] d x = t a n x - t a n^{- 1} x + C .$

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Find ∫(x2+sin2x)sec2x1+x2dx.

Let I=∫(x2+sin2x)sec2x1+x2dx.=∫(1+x2+2sin2x−1)sec2x1+x2dx ⇒I=∫(1+x2−cos2x)sec2x1+x2dx=∫(1+x2)sec2x−11+x2dx ⇒I=∫[sec2x−11+x2]dx=tan x−tan−1x+C.

Find $\int \frac{(x^{2} + s i n^{2} x) s e c^{2} x}{1 + x^{2}} d x$ .