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
Summary
This exercise in Chapter 19 Indefinite Integrals focuses on evaluating a variety of indefinite integrals using different integration techniques. The practice questions cover a wide range of functions, including polynomials, trigonometric functions, exponential functions, and rational functions.
Students will need to apply methods such as the power rule, integration by parts, substitution, and integration of trigonometric and rational functions to solve these problems. The aim is to help students develop proficiency in applying various integration techniques and strengthen their understanding of indefinite integrals.
The problems range in complexity, starting with simpler polynomial and trigonometric integrals, and progressing to more challenging integrals involving exponential and rational functions. Solving these practice questions will equip students with the skills to tackle a diverse set of indefinite integral problems.
