Advertisements
Advertisements
Question
\[\int\frac{\sin^6 x}{\cos^8 x} dx =\]
Options
tan 7x + C
- \[\frac{\tan^7 x}{7} + C\]
- \[\frac{\tan 7x}{7} + C\]
sec7 x + C
MCQ
Advertisements
Solution
\[\frac{\tan^7 x}{7} + C\]
\[\text{Let }I = \int\frac{\sin^6 x}{\cos^8 x}dx\]
\[ = \int\frac{\sin^6 x}{\cos^6 x} \times \frac{1}{\cos^2 x}dx\]
\[ = \int \tan^6 x \cdot \sec^2 x dx\]
\[\text{Putting }\tan x = t\]
\[ \Rightarrow \sec^2 x dx = dt\]
\[ \therefore I = \int t^6 \cdot dt\]
\[ = \frac{t^7}{7} + C\]
\[ = \frac{\tan^7 x}{7} + C ............\left( \because t = \tan x \right)\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int \left( 3x + 4 \right)^2 dx\]
\[\int\frac{2 x^4 + 7 x^3 + 6 x^2}{x^2 + 2x} dx\]
If f' (x) = 8x3 − 2x, f(2) = 8, find f(x)
\[\int\frac{1}{\sqrt{2x + 3} + \sqrt{2x - 3}} dx\]
` ∫ 1/ {1+ cos 3x} ` dx
\[\int\frac{x + 1}{\sqrt{2x + 3}} dx\]
\[\int\sqrt {e^x- 1} \text{dx}\]
\[\int \sin^5 x \cos x \text{ dx }\]
\[\int \sin^3 x \cos^6 x \text{ dx }\]
Evaluate the following integrals:
\[\int\frac{x^7}{\left( a^2 - x^2 \right)^5}dx\]
\[\int\frac{x}{x^4 + 2 x^2 + 3} dx\]
\[\int\frac{x}{\sqrt{x^4 + a^4}} dx\]
\[\int\frac{\sin 2x}{\sqrt{\sin^4 x + 4 \sin^2 x - 2}} dx\]
\[\int\frac{2x + 5}{\sqrt{x^2 + 2x + 5}} dx\]
\[\int\frac{3x + 1}{\sqrt{5 - 2x - x^2}} \text{ dx }\]
\[\int\frac{2x + 3}{\sqrt{x^2 + 4x + 5}} \text{ dx }\]
\[\int\frac{1}{4 \cos x - 1} \text{ dx }\]
\[\int\frac{1}{3 + 2 \sin x + \cos x} \text{ dx }\]
\[\int\frac{2 \tan x + 3}{3 \tan x + 4} \text{ dx }\]
\[\int\frac{8 \cot x + 1}{3 \cot x + 2} \text{ dx }\]
\[\int x \cos x\ dx\]
\[\int \sin^{- 1} \sqrt{\frac{x}{a + x}} \text{ dx }\]
\[\int e^x \left( \cot x + \log \sin x \right) dx\]
\[\int\frac{1}{\left( x - 1 \right) \left( x + 1 \right) \left( x + 2 \right)} dx\]
\[\int\frac{1}{x\left( x^n + 1 \right)} dx\]
\[\int\frac{x^2}{\left( x - 1 \right) \left( x + 1 \right)^2} dx\]
\[\int\frac{\cos x}{\left( 1 - \sin x \right)^3 \left( 2 + \sin x \right)} dx\]
\[\int\frac{1}{x^4 - 1} dx\]
Find \[\int\frac{2x}{\left( x^2 + 1 \right) \left( x^2 + 2 \right)^2}dx\]
\[\int\frac{x^2 + 1}{x^4 + x^2 + 1} \text{ dx }\]
\[\int e^x \left( 1 - \cot x + \cot^2 x \right) dx =\]
\[\int\frac{\sin^2 x}{\cos^4 x} dx =\]
\[\int \sin^5 x\ dx\]
\[\int\frac{1}{3 x^2 + 13x - 10} \text{ dx }\]
\[\int\frac{1}{a + b \tan x} \text{ dx }\]
\[\int\frac{1}{2 + \cos x} \text{ dx }\]
\[\int \sec^6 x\ dx\]
\[\int \sin^3 \left( 2x + 1 \right) \text{dx}\]
\[\int\frac{x + 3}{\left( x + 4 \right)^2} e^x dx =\]
