Advertisements
Advertisements
Question
\[\int\frac{\sin^3 x - \cos^3 x}{\sin^2 x \cos^2 x} dx\]
Sum
Solution
\[\int\left( \frac{\sin^3 x - \cos^3 x}{\sin^2 x \cdot \cos^2 x} \right)dx\]
\[ = \int\frac{\sin^3 x}{\sin^2 x \cdot \cos^2 x}dx - \int\frac{\cos^3 x}{\sin^2 x \cdot \cos^2 x}dx\]
\[ = \int\frac{\sin x}{\cos^2 x}dx - \int\frac{\cos x}{\sin^2 x}dx\]
\[ = \int\frac{\sin x}{\cos x} \times \frac{1}{\cos x}dx - \int\frac{\cos x}{\sin x} \times \frac{1}{\sin x}dx\]
`=∫ sec x tan x dx - ∫ "cosec" x cot x dx`
\[ = \sec x - \left( - \text{cosec x} \right) + C\]
\[ = \sec x + \text{cosec x }+ C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\left\{ \sqrt{x}\left( a x^2 + bx + c \right) \right\} dx\]
\[\int\frac{1}{1 - \sin x} dx\]
\[\int \tan^{- 1} \left( \frac{\sin 2x}{1 + \cos 2x} \right) dx\]
\[\int \left( a \tan x + b \cot x \right)^2 dx\]
If f' (x) = x + b, f(1) = 5, f(2) = 13, find f(x)
\[\int\frac{1}{\sqrt{x + 3} - \sqrt{x + 2}} dx\]
\[\int\frac{x^3}{x - 2} dx\]
\[\int\frac{x^2 + x + 5}{3x + 2} dx\]
\[\int\frac{3x + 5}{\sqrt{7x + 9}} dx\]
`∫ cos ^4 2x dx `
\[\int\frac{\cos x}{2 + 3 \sin x} dx\]
\[\int\frac{x \sin^{- 1} x^2}{\sqrt{1 - x^4}} dx\]
\[\ ∫ x \text{ e}^{x^2} dx\]
Evaluate the following integrals:
\[\int\cos\left\{ 2 \cot^{- 1} \sqrt{\frac{1 + x}{1 - x}} \right\}dx\]
\[\int\frac{x}{x^4 + 2 x^2 + 3} dx\]
\[\int\frac{1}{\sqrt{16 - 6x - x^2}} dx\]
\[\int\frac{x}{\sqrt{x^4 + a^4}} dx\]
\[\int\frac{2x - 3}{x^2 + 6x + 13} dx\]
\[\int\frac{x - 1}{3 x^2 - 4x + 3} dx\]
\[\int\frac{1}{4 \sin^2 x + 5 \cos^2 x} \text{ dx }\]
\[\int\frac{\sin 2x}{\sin^4 x + \cos^4 x} \text{ dx }\]
\[\int\frac{1}{1 - \tan x} \text{ dx }\]
\[\int\frac{\log x}{x^n}\text{ dx }\]
\[\int x \sin x \cos x\ dx\]
\[\int\left( x + 1 \right) \text{ e}^x \text{ log } \left( x e^x \right) dx\]
\[\int\left\{ \tan \left( \log x \right) + \sec^2 \left( \log x \right) \right\} dx\]
\[\int\frac{x^2 + x - 1}{x^2 + x - 6} dx\]
Evaluate the following integral:
\[\int\frac{x^2}{\left( x^2 + a^2 \right)\left( x^2 + b^2 \right)}dx\]
\[\int\frac{1}{x^4 + x^2 + 1} \text{ dx }\]
\[\int\frac{x^2 - 1}{x^4 + 1} \text{ dx }\]
\[\int\frac{1}{\left( 1 + x^2 \right) \sqrt{1 - x^2}} \text{ dx }\]
\[\int\frac{x}{4 + x^4} \text{ dx }\] is equal to
\[\int e^x \left( \frac{1 - \sin x}{1 - \cos x} \right) dx\]
\[\int\frac{x^9}{\left( 4 x^2 + 1 \right)^6}dx\] is equal to
\[\int\frac{\left( \sin^{- 1} x \right)^3}{\sqrt{1 - x^2}} \text{ dx }\]
\[\int\frac{1}{\sqrt{x^2 - a^2}} \text{ dx }\]
\[\int\frac{1}{\left( x^2 + 2 \right) \left( x^2 + 5 \right)} \text{ dx}\]
