Solved by AI teacher · Mathematics

∫ (sin³x + cos³x) / (sin²x cos²x) dx

Concept: Trigonometric Integral Simplification

Question

\int \frac{f(x) \pm g(x)}{h(x)} dx = \int \frac{f(x)}{h(x)} dx \pm \int \frac{g(x)}{h(x)} dx

Try it yourself first

Hint: Pehle, numerator ko denominator se alag-alag divide karke dekho, jaise solution mein step 1 mein kiya hai.

1
Split the fraction
Fraction split • Separate terms
$\frac{A+B}{C} = \frac{A}{C} + \frac{B}{C}$
2
Simplify first term
Cancel sin²x • sin x / cos²x • tan x sec x
$\frac{\sin^3x}{\sin^2x\cos^2x} = \frac{\sin x}{\cos^2x}$
3
Simplify second term
Cancel cos²x • cos x / sin²x • cot x cosec x
$\frac{\cos^3x}{\sin^2x\cos^2x} = \frac{\cos x}{\sin^2x}$
4
Integrate simplified terms
∫ tan x sec x dx = sec x • ∫ cot x cosec x dx = -cosec x • Add constant C
$\int \sec x \tan x dx = \sec x + C$

-cot x - tan x - 2x + C

Remember: Trigonometric identities se complex integrals simplify hote hain.

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Agar humein

\int \frac{\sin^2x + \cos^2x}{\sin^2x \cos^2x} dx

solve karna ho, toh pehla step kya hoga?

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