Find the best tutors and institutes for Class 12 Tuition
Q10:
Choose the correct answer in Exercises 9 and 10.
10. If $f(x) = \int_{0}^{x} t \sin t dt$, then $f'(x)$ is
Solution :
Step 1: Application of the Leibniz Integral Rule
To find the derivative of a function defined as a definite integral with variable limits, we employ the Leibniz Integral Rule [Fundamental Theorem of Calculus, Part 1]. For a function defined as:
$f(x) = \int_{g(x)}^{h(x)} \phi(t) dt$
The derivative $f'(x)$ is given by:
$f'(x) = \phi(h(x)) \cdot h'(x) - \phi(g(x)) \cdot g'(x)$
Step 2: Identification of Components
Given the function:
$f(x) = \int_{0}^{x} t \sin t \, dt$
- $\phi(t) = t \sin t$
- $h(x) = x$, therefore $h'(x) = \frac{d}{dx}(x) = 1$
- $g(x) = 0$, therefore $g'(x) = \frac{d}{dx}(0) = 0$
Step 3: Execution of the Differentiation
Applying the formula derived in Step 1:
$f'(x) = \phi(x) \cdot (1) - \phi(0) \cdot (0)$
$f'(x) = (x \sin x) \cdot 1 - 0$
$f'(x) = x \sin x$
Visual Representation of the Integrand
Step 4: Verification via Integration (Alternative Method)
If we evaluate the integral first using Integration by Parts ($\int u dv = uv - \int v du$):
Let $u = t$, $dv = \sin t \, dt \implies du = dt$, $v = -\cos t$
$\int t \sin t \, dt = -t \cos t - \int (-\cos t) \, dt = -t \cos t + \sin t$
Evaluating from $0$ to $x$:
$f(x) = [-t \cos t + \sin t]_{0}^{x} = (-x \cos x + \sin x) - (-0 \cos 0 + \sin 0) = \sin x - x \cos x$
Now, differentiate $f(x)$ with respect to $x$:
$f'(x) = \frac{d}{dx}(\sin x - x \cos x)$
$f'(x) = \cos x - [x(-\sin x) + \cos x(1)]$
$f'(x) = \cos x + x \sin x - \cos x$
$f'(x) = x \sin x$
Final Solution: $f'(x) = x \sin x$
More Questions from Class 12 Mathematics Integrals EXERCISE 7.9
- Q1: Evaluate the integrals in Exercises 1 to 8 using substitution. 1. $\int_{0}^{1} \frac{x}{x^2+1} dx$
- Q2: Evaluate the integrals in Exercises 1 to 8 using substitution. 2. $\int_{0}^{\frac{\pi}{2}} \sqrt{\sin \phi} \cos^5 \phi d\phi$
- Q3: Evaluate the integrals in Exercises 1 to 8 using substitution. 3. $\int_{0}^{1} \sin^{-1} (\frac{2x}{1+x^2}) dx$
- Q4: Evaluate the integrals in Exercises 1 to 8 using substitution. 4. $\int_{0}^{2} x \sqrt{x+2} dx$ (Put $x+2 = t^2$)
- Q5: Evaluate the integrals in Exercises 1 to 8 using substitution. 5. $\int_{0}^{\frac{\pi}{2}} \frac{\sin x}{1+\cos^2 x} dx$
- Q6: Evaluate the integrals in Exercises 1 to 8 using substitution. 6. $\int_{0}^{2} \frac{dx}{x+4-x^2}$
- Q7: Evaluate the integrals in Exercises 1 to 8 using substitution. 7. $\int_{-1}^{1} \frac{dx}{x^2+2x+5}$
- Q8: Evaluate the integrals in Exercises 1 to 8 using substitution. 8. $\int_{1}^{2} (\frac{1}{x} - \frac{1}{2x^2}) e^{2x} dx$
- Q9: Choose the correct answer in Exercises 9 and 10. 9. The value of the integral $\int_{\frac{1}{3}}^{1} \frac{(x-x^3)^{\frac{1}{3}}}{x^4} dx$ is
CBSE Solutions for Class 12 Mathematics Integrals
- Integrals Class 12 EXERCISE 7.1
- Integrals Class 12 EXERCISE 7.10
- Integrals Class 12 EXERCISE 7.2
- Integrals Class 12 EXERCISE 7.3
- Integrals Class 12 EXERCISE 7.4
- Integrals Class 12 EXERCISE 7.5
- Integrals Class 12 EXERCISE 7.6
- Integrals Class 12 EXERCISE 7.7
- Integrals Class 12 EXERCISE 7.8
- Integrals Class 12 Miscellaneous Exercise on Chapter 7
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Integrals
Top Tutor
Rajarajeshwari Nagar, Bangalore
12 yrs of
Exp
51 students
1960 hrs of classes completed
500
per hour
Online Classes
Student's Home
I have been training students for 12th Board examinations for more than a decade now.
He is outstanding as a teacher, knowledgeable and dedicated. He has helped my daughter so much with his knowledge, that the girl who was just running away from accounts is getting more than 90 percent
Top Tutor
Tri Nagar, Delhi
20 yrs of
Exp
4 students
32 hrs of classes completed
1000
per hour
Online Classes
I am Anshu Gupta. I was working at a reputed school in Delhi for 23 year before relocating to Singapore in 2021. I was teaching Physics to Grade 12 students. I have taught more than 3000 students in the last 24 years.I worked in Singapore as a freelance tutor for last 1.5 years I'm teaching Physics to the students of IB, IP, CBSE, CIE, ICSE, AP. I have a vast 24 year experience of teaching Physics to high school students I create individual based strategies for my students for best results I can create my own study material Good conceptual knowledge of my subject is my strength. I believe in creating a friendly and comfortable environment for free interaction. I have a lot of experience of online teaching ( 4 years ) So the online lessons are as effective as face to face lessons I'm teaching Physics to the students of IB(HL and SL), IP(H1 and H2), CBSE(11th and 12th), CIE(A and AS), ICSE, AP(Physics C). I am teaching 3 H2 and 3 IB students online at present
I craft this review as an offering of gratitude for the transformative influence she has on me. Please note that my review is completely transparent and true to my personal experience. I approached her with the request of covering Calculus from Class 12 RD Sharma (which extends to over 500 pages!). It's a challenge to find educators who truly prioritize depth and detail in their instruction and are willing to navigate the complexity that CBSE demands. However, under her guidance, we covered each exercise comprehensively, thoroughly and meticulously. Her teaching skills are truly top-notch and her sessions are sharply structured. Complex problems were masterfully explained. She is reservoir of knowledge, and always has all the answers to my doubts. Additionally, she keeps me engaged throughout the class, giving me questions to work through and think about independently. But it's not just her teaching that mesmerizes – it's her respect for punctuality that sets the tone for disciplined learning. Each class, every session, begins on time and makes it clear that she is a seasoned professional with kernels of experience to her name. All in all, if you're looking for an exceptional teacher, in fact the best on this platform, Deepa ma'am should be your only choice.
Super Tutor
Newada, Varanasi
17 yrs of
Exp
14 students
515 hrs of classes completed
1200
per hour
Online Classes
Tutor's home
Making chemistry simple & logical. 17+ years of experience teaching class 11, 12, jee & neet chemistry. B.Tech. (iit bhu), m.Tech. (iit delhi), ph.D. (iit bhu). Editor for the 13th edition of elementary problems in organic chemistry by m.S. Chouhan sir. Mentored students into top iits and aiims (bombay, kanpur, roorkee & more!). Top 10% global rank in the international teachers olympiad (2022 & 2023). Structured online classes with comprehensive notes & doubt-clearing. Let's master chemistry together!
I liked the way you taught. It was such a friendly manner. The way you are building up the tempo gradually is what I felt was good.
Top Tutor
Mahipalpur, Delhi
14 yrs of
Exp
2 hrs of classes completed
Student's Home
As an experienced teacher for 12 years, I have been teaching to make students interested in the subjects of Maths. My attitude is to help children improve in studies in the best possible way.
Top Tutor
Faridabad, Faridabad
5 yrs of
Exp
14 students
900
per hour
Online Classes
Student's Home
Tutor's home
I am teaching Physics from last 5 years. And within a week after getting classes from me, student falls in love with the subject. Eagerly waits for the next class.
He is an excellent teacher with a lot of knowledge, his is frank in nature,his teaching techniques are far better than other teachers, he keeps hi eye on each student of the batch, ravi sir is the best physics teacher I've ever seen.
