Advertisements
Advertisements
Question
\[\int \cos^5 x \text{ dx }\]
Sum
Advertisements
Solution
∫ cos5 x dx
= ∫ cos4 x . cos x dx
= ∫ (1 – sin2 x)2 cos x dx
Let sin x = t
⇒ cos x dx = dt
Now, ∫ (1 – sin2 x)2 cos x dx
= ∫ (1 – t2)2 . dt
= ∫ (1 + t4 – 2t2) dt
= ∫ dt + ∫ t4 dt – 2 ∫t2 dt
\[= t + \frac{t^5}{5} - \frac{2 t^3}{3} + C\]
\[ = \sin x + \frac{\sin^5 x}{5} - \frac{2}{3} \sin^3 x + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\left( \frac{m}{x} + \frac{x}{m} + m^x + x^m + mx \right) dx\]
\[\int\frac{1}{\left( 7x - 5 \right)^3} + \frac{1}{\sqrt{5x - 4}} dx\]
\[\int\frac{1}{\sqrt{x + 1} + \sqrt{x}} dx\]
\[\int\frac{2 - 3x}{\sqrt{1 + 3x}} dx\]
\[\int\frac{1 + \cot x}{x + \log \sin x} dx\]
\[\int\frac{\log\left( 1 + \frac{1}{x} \right)}{x \left( 1 + x \right)} dx\]
\[\int\frac{\cos x - \sin x}{1 + \sin 2x} dx\]
\[\int\frac{\cos\sqrt{x}}{\sqrt{x}} dx\]
\[\int\frac{x^2 + 3x + 1}{\left( x + 1 \right)^2} dx\]
\[\int\frac{x^4 + 1}{x^2 + 1} dx\]
\[\int\frac{1}{\sqrt{5 x^2 - 2x}} dx\]
\[\int\frac{x}{\sqrt{x^4 + a^4}} dx\]
\[\int\frac{x}{\sqrt{4 - x^4}} dx\]
\[\int\frac{x - 1}{3 x^2 - 4x + 3} dx\]
\[\int\frac{x^2 + 1}{x^2 - 5x + 6} dx\]
`int 1/(sin x - sqrt3 cos x) dx`
\[\int\frac{1}{1 - \tan x} \text{ dx }\]
\[\int \log_{10} x\ dx\]
\[\int \sec^{- 1} \sqrt{x}\ dx\]
\[\int x\left( \frac{\sec 2x - 1}{\sec 2x + 1} \right) dx\]
\[\int e^x \left( \frac{\sin x \cos x - 1}{\sin^2 x} \right) dx\]
\[\int\left( x + 1 \right) \sqrt{x^2 + x + 1} \text{ dx }\]
\[\int\frac{2x + 1}{\left( x + 1 \right) \left( x - 2 \right)} dx\]
\[\int\frac{x^2}{\left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)} dx\]
\[\int\frac{x^3}{\left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)} dx\]
\[\int\frac{x}{\left( x^2 - a^2 \right) \left( x^2 - b^2 \right)} dx\]
\[\int\frac{x^4}{\left( x - 1 \right) \left( x^2 + 1 \right)} dx\]
\[\int\frac{1}{x^4 + x^2 + 1} \text{ dx }\]
\[\int\frac{1}{\left( x - 1 \right) \sqrt{2x + 3}} \text{ dx }\]
\[\int\frac{x}{\left( x - 3 \right) \sqrt{x + 1}} \text{ dx}\]
\[\int \sin^4 2x\ dx\]
\[\int\frac{1}{e^x + 1} \text{ dx }\]
\[\int x \sin^5 x^2 \cos x^2 dx\]
\[\int\frac{\sin 2x}{\sin^4 x + \cos^4 x} \text{ dx }\]
\[\int\frac{1 + \sin x}{\sin x \left( 1 + \cos x \right)} \text{ dx }\]
\[\int \tan^{- 1} \sqrt{\frac{1 - x}{1 + x}} \text{ dx }\]
\[\int \sin^{- 1} \sqrt{\frac{x}{a + x}} \text{ dx}\]
\[\int\frac{x}{x^3 - 1} \text{ dx}\]
Find: `int (3x +5)/(x^2+3x-18)dx.`
