Advertisements
Advertisements
Question
` ∫ cos 3x cos 4x` dx
Sum
Solution
\[\int\text{cos 4x }\text{cos 3x dx}\]
` = 1/2 ∫ 2 cos 4x cos 3x dx `
\[ = \frac{1}{2}\int\left[ \text{cos} \left( 4x + 3x \right) + \text{cos }\left( 4x - 3x \right) \right]dx \left[ \therefore \text{2 }\text{cos A }\text{cos B} = \text{cos} \left( A + B \right) + \text{cos }\left( A - B \right) \right]\]
\[ = \frac{1}{2}\int\left( \text{cos} \left( 7x \right) + \text{cos x} \right) dx\]
\[ = \frac{1}{2}\left[ \frac{\sin 7x}{7} + \sin x \right] + C\]
\[ = \frac{1}{14}\sin 7x + \frac{1}{2}\sin x + C\]
\[ = \frac{1}{2}\int\left[ \text{cos} \left( 4x + 3x \right) + \text{cos }\left( 4x - 3x \right) \right]dx \left[ \therefore \text{2 }\text{cos A }\text{cos B} = \text{cos} \left( A + B \right) + \text{cos }\left( A - B \right) \right]\]
\[ = \frac{1}{2}\int\left( \text{cos} \left( 7x \right) + \text{cos x} \right) dx\]
\[ = \frac{1}{2}\left[ \frac{\sin 7x}{7} + \sin x \right] + C\]
\[ = \frac{1}{14}\sin 7x + \frac{1}{2}\sin x + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\frac{5 \cos^3 x + 6 \sin^3 x}{2 \sin^2 x \cos^2 x} dx\]
\[\int\frac{1}{\sqrt{x + 3} - \sqrt{x + 2}} dx\]
\[\int\frac{x^3}{x - 2} dx\]
\[\int\left( 5x + 3 \right) \sqrt{2x - 1} dx\]
\[\int\sqrt{\frac{1 - \sin 2x}{1 + \sin 2x}} dx\]
\[\int\frac{\sin 2x}{\sin 5x \sin 3x} dx\]
\[\int\left( \frac{x + 1}{x} \right) \left( x + \log x \right)^2 dx\]
\[\int\frac{\sin \left( \text{log x} \right)}{x} dx\]
` ∫ x {tan^{- 1} x^2}/{1 + x^4} dx`
\[\int\frac{\left( \sin^{- 1} x \right)^3}{\sqrt{1 - x^2}} dx\]
\[\int\frac{x + \sqrt{x + 1}}{x + 2} dx\]
` ∫ tan x sec^4 x dx `
\[\int \sin^5 x \text{ dx }\]
\[\int \cos^5 x \text{ dx }\]
\[\int\frac{1}{\sin^4 x \cos^2 x} dx\]
\[\int\frac{dx}{e^x + e^{- x}}\]
\[\int\frac{2x}{2 + x - x^2} \text{ dx }\]
\[\int\frac{x + 2}{\sqrt{x^2 + 2x - 1}} \text{ dx }\]
\[\int\frac{1}{4 \cos^2 x + 9 \sin^2 x}\text{ dx }\]
`int"x"^"n"."log" "x" "dx"`
` ∫ x tan ^2 x dx
\[\int \tan^{- 1} \sqrt{\frac{1 - x}{1 + x}} dx\]
\[\int\frac{\sqrt{16 + \left( \log x \right)^2}}{x} \text{ dx}\]
\[\int\left( 2x + 3 \right) \sqrt{x^2 + 4x + 3} \text{ dx }\]
\[\int\frac{2x - 3}{\left( x^2 - 1 \right) \left( 2x + 3 \right)} dx\]
\[\int\frac{1}{\left( x - 1 \right) \left( x + 1 \right) \left( x + 2 \right)} dx\]
\[\int\frac{x^2}{\left( x - 1 \right) \left( x + 1 \right)^2} dx\]
\[\int\frac{18}{\left( x + 2 \right) \left( x^2 + 4 \right)} dx\]
\[\int\frac{3x + 5}{x^3 - x^2 - x + 1} dx\]
\[\int\frac{1}{\cos x \left( 5 - 4 \sin x \right)} dx\]
\[\int\frac{x}{\left( x - 3 \right) \sqrt{x + 1}} \text{ dx}\]
\[\int\frac{1}{7 + 5 \cos x} dx =\]
\[\int\frac{x^4 + x^2 - 1}{x^2 + 1} \text{ dx}\]
\[\int \text{cosec}^2 x \text{ cos}^2 \text{ 2x dx} \]
\[\int\frac{5x + 7}{\sqrt{\left( x - 5 \right) \left( x - 4 \right)}} \text{ dx }\]
\[\int\frac{1}{\sin^2 x + \sin 2x} \text{ dx }\]
\[\int\frac{x^3}{\sqrt{x^8 + 4}} \text{ dx }\]
\[\int\frac{1}{1 + 2 \cos x} \text{ dx }\]
\[\int\frac{x^2}{\left( x - 1 \right)^3 \left( x + 1 \right)} \text{ dx}\]
\[\int \sin^3 \left( 2x + 1 \right) \text{dx}\]
