Advertisements
Advertisements
Question
\[\int\frac{e^x}{\sqrt{16 - e^{2x}}} dx\]
Sum
Advertisements
Solution
\[\int\frac{e^x dx}{\sqrt{16 - \left( e^x \right)^2}}\]
\[\text{ let } e^x = t\]
\[ \Rightarrow e^x dx = dt\]
\[Now, \int\frac{e^x dx}{\sqrt{16 - \left( e^x \right)^2}}\]
\[ = \int\frac{dt}{\sqrt{16 - t^2}}\]
\[ = \int\frac{dt}{\sqrt{4^2 - t^2}}\]
\[ = \sin^{- 1} \left( \frac{t}{4} \right) + C\]
\[ = \sin^{- 1} \left( \frac{e^x}{4} \right) + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\frac{x^6 + 1}{x^2 + 1} dx\]
\[\int\frac{\left( 1 + \sqrt{x} \right)^2}{\sqrt{x}} dx\]
\[\int \left( 3x + 4 \right)^2 dx\]
\[\int\frac{1 - \cos 2x}{1 + \cos 2x} dx\]
` ∫ 1/ {1+ cos 3x} ` dx
\[\int\frac{2x + 3}{\left( x - 1 \right)^2} dx\]
\[\int\frac{3x + 5}{\sqrt{7x + 9}} dx\]
\[\int \sin^2\text{ b x dx}\]
\[\int\frac{\cos x}{\cos \left( x - a \right)} dx\]
\[\int\frac{\sec x \tan x}{3 \sec x + 5} dx\]
\[\int\frac{a}{b + c e^x} dx\]
\[\int\left\{ 1 + \tan x \tan \left( x + \theta \right) \right\} dx\]
\[\int\left( \frac{x + 1}{x} \right) \left( x + \log x \right)^2 dx\]
` ∫ x {tan^{- 1} x^2}/{1 + x^4} dx`
\[\int \cos^5 x \text{ dx }\]
\[\int\frac{1}{x^2 - 10x + 34} dx\]
\[\int\frac{\cos x}{\sin^2 x + 4 \sin x + 5} dx\]
\[\int\frac{\sin 2x}{\sqrt{\sin^4 x + 4 \sin^2 x - 2}} dx\]
\[\int\frac{1}{\sqrt{\left( 1 - x^2 \right)\left\{ 9 + \left( \sin^{- 1} x \right)^2 \right\}}} dx\]
\[\int\frac{x + 1}{x^2 + x + 3} dx\]
\[\int\frac{a x^3 + bx}{x^4 + c^2} dx\]
\[\int\frac{x^2 + 1}{x^2 - 5x + 6} dx\]
\[\int\frac{1}{3 + 4 \cot x} dx\]
\[\int\left( \frac{1}{\log x} - \frac{1}{\left( \log x \right)^2} \right) dx\]
\[\int\left( x + 1 \right) \sqrt{x^2 - x + 1} \text{ dx}\]
\[\int\frac{x^2 + x - 1}{x^2 + x - 6} dx\]
\[\int\frac{1}{\left( x - 1 \right) \left( x + 1 \right) \left( x + 2 \right)} dx\]
\[\int\frac{x^2 + 1}{\left( x - 2 \right)^2 \left( x + 3 \right)} dx\]
\[\int\frac{x^2 + x - 1}{\left( x + 1 \right)^2 \left( x + 2 \right)} dx\]
The primitive of the function \[f\left( x \right) = \left( 1 - \frac{1}{x^2} \right) a^{x + \frac{1}{x}} , a > 0\text{ is}\]
\[\int \text{cosec}^2 x \text{ cos}^2 \text{ 2x dx} \]
\[\int \sin^4 2x\ dx\]
\[\int\frac{1}{e^x + e^{- x}} dx\]
\[\int\sin x \sin 2x \text{ sin 3x dx }\]
\[\int\frac{\sin^6 x}{\cos x} \text{ dx }\]
\[\int\frac{\sin^2 x}{\cos^6 x} \text{ dx }\]
\[\int\frac{\sqrt{1 - \sin x}}{1 + \cos x} e^{- x/2} \text{ dx}\]
\[\int\frac{\cot x + \cot^3 x}{1 + \cot^3 x} \text{ dx}\]
