Advertisements
Advertisements
Question
\[\int e^x \left( \cos x - \sin x \right) dx\]
Sum
Advertisements
Solution
\[\text{ Let I } = \int e^x \left( \cos x - \sin x \right) dx \]
\[\text{ let e}^x \cos x = t \]
\[\text{ Diff both sides w . r . t x}\]
\[ e^x \cdot \cos x + e^x \left( - \sin x \right) = \frac{dt}{dx} \text{ Put e}^x f\left( x \right) = t\]
\[ \Rightarrow e^x \left( \cos x - \sin x \right) dx = dt\]
\[ \therefore \int e^x \left( \cos x - \sin x \right) dx = \int dt\]
\[ \Rightarrow I = t + C\]
\[ = e^x \cos x + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\sqrt{x}\left( x^3 - \frac{2}{x} \right) dx\]
\[\int \left( \tan x + \cot x \right)^2 dx\]
\[\int \cot^{- 1} \left( \frac{\sin 2x}{1 - \cos 2x} \right) dx\]
If f' (x) = 8x3 − 2x, f(2) = 8, find f(x)
\[\int\frac{2x + 3}{\left( x - 1 \right)^2} dx\]
` ∫ sin 4x cos 7x dx `
\[\int\frac{e^x + 1}{e^x + x} dx\]
\[\int\left( \frac{x + 1}{x} \right) \left( x + \log x \right)^2 dx\]
\[\int\frac{2x - 1}{\left( x - 1 \right)^2} dx\]
\[\int\frac{1}{\sqrt{\left( 2 - x \right)^2 - 1}} dx\]
\[\int\frac{1}{\sqrt{5 - 4x - 2 x^2}} dx\]
\[\int\frac{\sin 2x}{\sqrt{\sin^4 x + 4 \sin^2 x - 2}} dx\]
\[\int\frac{1}{\sqrt{\left( 1 - x^2 \right)\left\{ 9 + \left( \sin^{- 1} x \right)^2 \right\}}} dx\]
\[\int\frac{1}{3 + 2 \cos^2 x} \text{ dx }\]
\[\int x e^x \text{ dx }\]
\[\int x \sin x \cos x\ dx\]
\[\int x \sin x \cos 2x\ dx\]
\[\int\frac{x^3 \sin^{- 1} x^2}{\sqrt{1 - x^4}} \text{ dx }\]
\[\int e^x \left( \tan x - \log \cos x \right) dx\]
\[\int e^x \left( \frac{x - 1}{2 x^2} \right) dx\]
∴\[\int e^{2x} \left( - \sin x + 2 \cos x \right) dx\]
\[\int\left( x + 1 \right) \sqrt{x^2 - x + 1} \text{ dx}\]
\[\int\frac{3 + 4x - x^2}{\left( x + 2 \right) \left( x - 1 \right)} dx\]
\[\int\frac{x^2}{\left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)} dx\]
\[\int\frac{x^3}{\left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)} dx\]
\[\int\frac{1}{x\left( x^n + 1 \right)} dx\]
\[\int\frac{x^2 + 1}{\left( x - 2 \right)^2 \left( x + 3 \right)} dx\]
\[\int\frac{1}{\left( x - 1 \right) \sqrt{2x + 3}} \text{ dx }\]
\[\int\frac{x}{\left( x - 3 \right) \sqrt{x + 1}} \text{ dx}\]
\[\int\frac{x}{\left( x^2 + 2x + 2 \right) \sqrt{x + 1}} \text{ dx}\]
If \[\int\frac{1}{5 + 4 \sin x} dx = A \tan^{- 1} \left( B \tan\frac{x}{2} + \frac{4}{3} \right) + C,\] then
\[\int \cos^3 (3x)\ dx\]
\[\int\frac{1}{e^x + 1} \text{ dx }\]
\[\int\frac{\sin 2x}{\sin^4 x + \cos^4 x} \text{ dx }\]
\[\int\frac{1}{\sqrt{3 - 2x - x^2}} \text{ dx}\]
\[\int\sqrt{\frac{1 + x}{x}} \text{ dx }\]
\[\int \cos^{- 1} \left( 1 - 2 x^2 \right) \text{ dx }\]
\[\int\sqrt{\frac{1 - \sqrt{x}}{1 + \sqrt{x}}} \text{ dx}\]
\[\int\frac{\cos^7 x}{\sin x} dx\]
