What Integrated courses one can pursue after completing class 10th as me and my friends want to take admission in it?

According to my knowledge there is no integrated course offered 10th by any institute in India. The integrated courses can be enrolled after 12th that ranges from four to five years of programs including theory and practical semesters and after passing with these integrated degree programs after 12th class one can join with doctoral programs.

Some of the institutes that offer integrated degree programs after 12th class:

Sharda University
NMIMS
Punjab University
BITS, Pilani
Pondicherry University
Osmania University
Nirma University
SOA University
Indian Institute of Technology
Himalayan University

Some of the courses offered are as follows:

Biochemical Engineering and Biotechnology
Computer Science and Engineering
B.Tech in Chemical Engineering and M.Tech in Computer Apps in Chemical
Engineering B.Tech in Chemical Engineering and M.Tech in Process Engineering and Design
B.Tech in Electrical Engineering and M.Tech in Information and Comm Tech
Msc-PhD (Operations Research)
5-yr Dual Degree (M.Tech) in Biological Engineering
BA LLB, BBA LLB & B.Com LLB
Integrated (BBA + MBA)