Integrate the function- - e x sin x -cos x dx

Let I =∫ e^x (sin x+cos x)dx. Let f(x)=sin x ⇒ f'(x)=cos x ⇒ then, I =∫ e^x [f(x)+f'(x)]dx We know that ∫ e^x (f(x)+f'(x))dx =e^x f(x) ∴ I =e^x sin. x+C ...

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Standard XII

Mathematics

Standard Formulae - 1

Integrate the...

Question

Integrate the function.
$\int e^{x} (s i n x + c o s x) d x .$

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Solution

Let $I = \int e^{x} (s i n x + c o s x) d x .$
Let $f (x) = s i n x \Rightarrow f^{'} (x) = c o s x \Rightarrow t h e n, I = \int e^{x} [f (x) + f^{'} (x)] d x$
We know that $\int e^{x} (f (x) + f^{'} (x)) d x = e^{x} f (x) ∴ I = e^{x} s i n . x + C$

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Integrate the function. ∫ex(sinx+cosx)dx.

Let I=∫ex(sinx+cosx)dx. Let f(x)=sinx⇒f′(x)=cosx⇒then,I=∫ex[f(x)+f′(x)]dx We know that ∫ex(f(x)+f′(x))dx=exf(x)∴I=exsin.x+C

Integrate the function.
$\int e^{x} (s i n x + c o s x) d x .$

Let $I = \int e^{x} (s i n x + c o s x) d x .$
Let $f (x) = s i n x \Rightarrow f^{'} (x) = c o s x \Rightarrow t h e n, I = \int e^{x} [f (x) + f^{'} (x)] d x$
We know that $\int e^{x} (f (x) + f^{'} (x)) d x = e^{x} f (x) ∴ I = e^{x} s i n . x + C$