Question
Integrate the following integrals:
$\int\sin4\text{x}\cos7\text{x dx}$
$\int\sin4\text{x}\cos7\text{x dx}$
✓
Answer
$\int\sin4\text{x}\cos7\text{x dx}$
$=\frac{1}{2}\int2\cos7\text{x}\sin4\text{x dx}$
$=\frac{1}{2}\int\big[\sin(7\text{x}+4\text{x})-\sin(7\text{x}-4\text{x})\big]\text{dx}$ $[\therefore2\cos\text{A}\sin\text{B}=\sin(\text{A}+\text{B})-\sin(\text{A}-\text{B})\big]$
$=\frac{1}{2}\int\big(\sin(11\text{x})-\sin(3\text{x})\big)\text{dx}$
$=\frac{1}{2}\Big[-\frac{\cos(11\text{x})}{11}+\frac{\cos(3\text{x})}{3}\Big]+\text{c}$
$=-\frac{\cos(11\text{x})}{22}+\frac{\cos(3\text{x})}{6}$
$=\frac{1}{2}\int2\cos7\text{x}\sin4\text{x dx}$
$=\frac{1}{2}\int\big[\sin(7\text{x}+4\text{x})-\sin(7\text{x}-4\text{x})\big]\text{dx}$ $[\therefore2\cos\text{A}\sin\text{B}=\sin(\text{A}+\text{B})-\sin(\text{A}-\text{B})\big]$
$=\frac{1}{2}\int\big(\sin(11\text{x})-\sin(3\text{x})\big)\text{dx}$
$=\frac{1}{2}\Big[-\frac{\cos(11\text{x})}{11}+\frac{\cos(3\text{x})}{3}\Big]+\text{c}$
$=-\frac{\cos(11\text{x})}{22}+\frac{\cos(3\text{x})}{6}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Evaluate the following integrals:
$\int\sqrt{16\text{x}^2+25}\text{dx}$
→$\int\sqrt{16\text{x}^2+25}\text{dx}$
Evaluate the following integrals:
$\int\cot^{\text{n}}\text{cosec}^2\text{x}\text{ dx},\text{ n}\neq-1$
→$\int\cot^{\text{n}}\text{cosec}^2\text{x}\text{ dx},\text{ n}\neq-1$
Find the equation of the plane passing through the point $(7,8,6)$ and parallel to the plane $\overline{ r } \cdot(6 \hat{ i }+8 \hat{ j }+7 \widehat{ k })=0$
→Find the joint equation of the pair of lines passing through the origin which are perpendicular respectively to the lines represented by $5 x^2+2 x y-3 y^2=0$.
→Diffrentiate the following w.r.t.x
→$\frac{1+\sin x^{\circ}}{1-\sin x^{\circ}}$
If $\text{A}=\begin{bmatrix}1&2\\0&3 \end{bmatrix}$ is written as B + C, where B is a symmetric matrix and C is a skew- symmetric matrix, then B is equal to.
→Find the intervals in which the following functions are increasing or decreasing.
$f(x) = 6+ 12x + 3x^2 - 2x^3$
→$f(x) = 6+ 12x + 3x^2 - 2x^3$
Express the following circuits in the symbolic form of logic and writ the input-output table.
Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
$\frac{\text{x}^2}{\text{a}^2}+\frac{\text{y}^2}{\text{b}^2}=1$
→$\frac{\text{x}^2}{\text{a}^2}+\frac{\text{y}^2}{\text{b}^2}=1$
With usual notations prove that $2\left\{a \sin ^2 \frac{C}{2}+c \sin ^2 \frac{A}{2}\right\}=a-b+c$.
→