Class 12 RD Sharma Solutions- Chapter 19 Indefinite Integrals – Exercise 19.7

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Integrate the following integrals:

Question 1. ∫sin4x cos7x dx

Solution:

Let I= $\int \sin4x\cos7x\,dx$

We know,

$2\sin A \cos B= \sin(A+B)+\sin(A-B)$

Applying this formula to the given question we get

I= $\int \frac 1 2(\sin(4x+7x)+\sin(4x-7x))\,dx$

= $\int \frac 1 2(\sin11x+\sin(-3x)\,dx$

= $\int\frac 1 2 (\sin 11x -\sin3x)\,dx$ $[\because \sin(-\theta)=-\sin\theta]$

= $\frac 1 2(\int\sin 11x -\int\sin3x)\,dx$

We know,

$\int \sin ax\,dx=-\frac 1 a\cos ax+C$

Applying this formula to the given question we get

I= $\frac 1 2(-\frac {1} {11} \cos 11x +\frac 1 3 \cos3x)$

$\therefore$ I= $-\frac {1} {22} \cos 11x +\frac 1 6 \cos3x+C$

Question 2. ∫ cos3x cos4x dx

Solution:

Let I= $\int \cos3x\cos4x\,dx$

Multiplying and dividing the equation by 2,we get

I= $\frac 1 2\int 2\cos3x\cos4x\,dx$

We know,

$2\cos A \cos B= \cos(A+B)+\cos(A-B)$

Applying this formula to the given question we get

I= $\frac 1 2\int( \cos(3x+4x)+\cos(3x-4x))\,dx$

= $\frac 1 2\int( \cos 7x+\cos x)\,dx$ $[\because \cos(-\theta)=\cos\theta]$

We know,

$\int \cos x\,dx=\sin x+C$ and $\int \cos ax\,dx=\frac 1 a\sin ax+C$

Applying these formulas to the given question we get

I= $\frac 1 2 (\frac {\sin 7x} 7+\sin x) + C$

$\therefore$ I= $(\frac {\sin 7x} {14}+\frac {\sin x} 2) + C$

Question 3. ∫ cosmx cosnx dx, m≠n

Solution:

Let I= $\int \cos mx \cos nx\,dx , m \ne n$

Multiplying and dividing the equation by 2,we get

I= $\frac 1 2\int 2\cos mx\cos nx\,dx$

We know,

$2\cos A \cos B= \cos(A+B)+\cos(A-B)$

Applying this formula to the given question we get

I= $\frac 1 2\int( \cos(m+n)x+\cos(m-n)x)\,dx$

We know,

$\int \cos ax\,dx=\frac 1 a\sin ax+C$

Applying these formulas to the given question we get

$\therefore$ I= $\frac 1 2 (\frac {\sin (m+n)x} {m+n}+\frac {\sin (m-n)x} {m-n}) + C$

Question 4. ∫ sinmx cosnx dx, m≠n

Solution:

Let I= $\int \sin mx \cos nx\,dx , m \ne n$

Multiplying and dividing the equation by 2,we get

I= $\frac 1 2\int 2\sin mx\cos nx\,dx$

We know,

$2\sin A \cos B= \sin(A+B)+\sin(A-B)$

Applying this formula to the given question we get

I= $\frac 1 2\int( \sin(m+n)x+\sin(m-n)x)\,dx$

We know,

$\int \sin ax\,dx=-\frac 1 a\cos ax+C$

Applying these formulas to the given question we get

$\therefore$ I= $\frac 1 2 (-\frac {\cos (m+n)x} {m+n}-\frac {\cos (m-n)x} {m-n}) + C$

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Last Updated : 13 Jan, 2021

Class 12 RD Sharma Solutions- Chapter 19 Indefinite Integrals - Exercise 19.6

Class 12 RD Sharma Solutions - Chapter 19 Indefinite Integrals - Exercise 19.8 | Set 1

Chapter 1: Relations

Chapter 2: Functions

Chapter 3: Binary Operations

Chapter 4: Inverse Trigonometric Functions

Chapter 5: Algebra of Matrices

Chapter 6: Determinants

Chapter 7: Adjoint and Inverse of a Matrix

Chapter 8: Solutions of Simultaneous Linear Equations

Chapter 9: Continuity

Chapter 10: Differentiability

Chapter 11: Differentiation

Chapter 12: Higher Order Derivatives

Chapter 13: Derivative as a Rate Measurer

Chapter 14: Differentials, Errors, and Approximations

Chapter 15: Mean Value Theorems

Chapter 16: Tangents and Normals

Chapter 17: Increasing and Decreasing Functions

Chapter 18: Maxima and Minima

Chapter 19: Indefinite Integrals

Chapter 20: Definite Integrals

Chapter 21: Areas of Bounded Regions

Chapter 22: Differential Equations

Chapter 23: Algebra of Vectors

Chapter 24: Scalar Or Dot Product

Chapter 25: Vector or Cross Product

Chapter 26: Scalar Triple Product

Chapter 27: Direction Cosines and Direction Ratios

Chapter 28: Straight Line in Space

Chapter 29: The Plane

Chapter 30: Linear programming

Chapter 31: Probability

Chapter 32: Mean and Variance of a Random Variable

Chapter 33: Binomial Distribution

Chapter 1: Relations

Chapter 2: Functions

Chapter 3: Binary Operations

Chapter 4: Inverse Trigonometric Functions

Chapter 5: Algebra of Matrices

Chapter 6: Determinants

Chapter 7: Adjoint and Inverse of a Matrix

Chapter 8: Solutions of Simultaneous Linear Equations

Chapter 9: Continuity

Chapter 10: Differentiability

Chapter 11: Differentiation

Chapter 12: Higher Order Derivatives

Chapter 13: Derivative as a Rate Measurer

Chapter 14: Differentials, Errors, and Approximations

Chapter 15: Mean Value Theorems

Chapter 16: Tangents and Normals

Chapter 17: Increasing and Decreasing Functions

Chapter 18: Maxima and Minima

Chapter 19: Indefinite Integrals

Chapter 20: Definite Integrals

Chapter 21: Areas of Bounded Regions

Chapter 22: Differential Equations

Chapter 23: Algebra of Vectors

Chapter 24: Scalar Or Dot Product

Chapter 25: Vector or Cross Product

Chapter 26: Scalar Triple Product

Chapter 27: Direction Cosines and Direction Ratios

Chapter 28: Straight Line in Space

Chapter 29: The Plane

Chapter 30: Linear programming

Chapter 31: Probability

Chapter 32: Mean and Variance of a Random Variable

Chapter 33: Binomial Distribution

Class 12 RD Sharma Solutions- Chapter 19 Indefinite Integrals – Exercise 19.7

Integrate the following integrals:

Question 1. ∫sin4x cos7x dx

Question 2. ∫ cos3x cos4x dx

Question 3. ∫ cosmx cosnx dx, m≠​n

Question 4. ∫ sinmx cosnx dx, m≠n

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Question 3. ∫ cosmx cosnx dx, m≠n