Mathematics

# If an antiderivative of $\displaystyle f ( x )$ is $e ^ { x }$ and that of $\displaystyle g ( x )$ is $\cos x ,$ then $\int f ( x ) \cos x d x + \int g ( x ) e ^ { x } d x =$

$e ^ { x } \cos x + c$

##### SOLUTION
$\displaystyle\int \underset{I}{f(x)}\underset{II}{\cos x}dx+\displaystyle\int \underset{I}{g(x)}\underset{II}{e^x}d(x)$
Using by parts
$f(x)\displaystyle\int \cos xdx-\displaystyle\int f'(x)\cdot \sin xdx+g(x)\displaystyle\int e^x-\displaystyle\int g'(x)e^x$
$\displaystyle\int \underset{II}{f(x)}\underset{I}{\cos x}dx+\displaystyle\underset{II}{g(x)}\underset{I}{e^x}dx$
Using by parts
$\cos x\displaystyle\int f(x)+\displaystyle\int \sin x(\displaystyle\int f(x)dx)dx+e^x\displaystyle\int g(x)-\displaystyle\int e^x(\displaystyle\int g(x)dx)$
$=\cos xe^x+\displaystyle\int \sin x e^xdx+e^x\cos x-\displaystyle\int e^x\cos x dx$
$=2\cos x e^x+\displaystyle\int \sin x e^x dx+\displaystyle\int e^x\cos x-\displaystyle\int e^x\cos x dx$
$=2\cos x e^x+\displaystyle\int \sin x e^xdx+\displaystyle\int \underset{II}{e^x}\underset{I}{(-\cos x)}dx$
$=2\cos x e^x+\displaystyle\int \sin x e^xdx+(-\cos x)e^x-\displaystyle\int \sin x e^x dx$
$=\cos x e^x+c$.

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Subjective Medium
Solve : $\displaystyle \int \sin^{-1} x . \dfrac{1}{x^2} dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
Primitive of $f(x)=x.2^{\ln(x^2+1)}$ w.r.t  $x$  is
• A. $\displaystyle \frac{2^{\ln(x^2+1)}}{2(x^2+1)}+C$
• B. $\displaystyle \frac{(x^2+1)2^{\ln(x^2+1)}}{\ln2+1}+C$
• C. $\displaystyle \frac {(x^2+1)^{\ln2}}{2(\ln 2+1)}+C$
• D. $\displaystyle \frac{(x^2+1)^{(\ln2+1)}}{2(\ln 2+1)}+C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Resolve $\displaystyle \frac{2x^2-11x+5}{(x-3)(x^2+2x+5)}$ into partial fractions.
• A. $\displaystyle \frac{1}{2(x-3)} - \frac{(5x-5)}{2(x^2+2x+5)}$
• B. $\displaystyle \frac{1}{(x-3)} + \frac{(5x-5)}{(x^2+2x+5)}$
• C. $\displaystyle \frac{1}{2(x+3)} + \frac{(5x-5)}{2(x^2+2x+5)}$
• D. $\displaystyle \frac{1}{2(x-3)} + \frac{(5x-5)}{2(x^2+2x+5)}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium

Evaluate:

$\displaystyle \int {\dfrac{{1 - {x^3}}}{{\left( {1 - 2x} \right)}}} \,dx$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
If $f(x)={ x }^{ 3 }+x,$then $\int _{ 1 }^{ 2 }{ f\left( x \right) dx+2\int _{ 1 }^{ 5 }{ { f }^{ -1 }\left( 2x \right) dx } }$
• A. 9
• B. 8
• C. 18
• D. 21