Question

$A:\int e^{3x}\cos 4xdx=\frac{e^{3x}}{25}(3\cos 4x+4\sin 4x)+c$.
$R:\int e^{ax}\cos (bx+c)dx=\frac{e^{ax}}{a^2+b^2}$ $\left[a\cos \right(bx+c)+b\sin (bx+c\left)\right]+k$

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true but R is not correct explanation of A
• C. A is true but R is false
• D. A is false but R is true.

Answer 1

The correct option is A Both A and R are true and R is the correct explanation of A

We know, from the property of integrals,

$\int e^{ax}\cos (bx+c)=\frac{e^{ax}}{a^2+b^2}\times \left[a\cos \right(bx+c)+b\sin (bx+c\left)\right]+k$

Thus substituting the values in the identity, with a = 3 b = 4 and c = 0

$\int e^{3x}\cos \left(4x\right)=\frac{e^{3x}}{3^2+4^2}\times \left[3\cos \right(4x)+4\sin (4x\left)\right]+k$

= $\frac{e^{3x}}{25}\times \left[3cos\right(4x)+4sin(4x\left)\right]+k$

Hence, both A and R are true and R is the correct explanation of A.