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An unordered shopping list of your favorite foods
An ordered list of todo’s you need to get done today
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Mathematics Questions

1. The integral $\int \frac{1}{\sqrt[4]{(x-1)^3 (x+2)^5}} \, dx$ is equal to (where $C$ is a constant of integration):

(A) $\dfrac{3}{4} \left( \dfrac{x+2}{x-1} \right)^{\frac{1}{4}} + C$

(B) $\dfrac{3}{4} \left( \dfrac{x+2}{x-1} \right)^{\frac{5}{4}} + C$

(D) $\dfrac{4}{3} \left( \dfrac{x-1}{x+2} \right)^{\frac{5}{4}} + C$

Click to View Answer

Correct Answer: [Insert Correct Option]

Mathematics Questions

1. The integral $\int \frac{1}{\sqrt[4]{(x-1)^3 (x+2)^5}} \, dx$ is equal to (where $C$ is a constant of integration):

(A) $\dfrac{3}{4} \left( \dfrac{x+2}{x-1} \right)^{\frac{1}{4}} + C$

(B) $\dfrac{3}{4} \left( \dfrac{x+2}{x-1} \right)^{\frac{5}{4}} + C$

(D) $\dfrac{4}{3} \left( \dfrac{x-1}{x+2} \right)^{\frac{5}{4}} + C$

Click to View Answer

Correct Answer: [Insert Correct Option]

Mathematics Questions

1. The integral $\int \frac{1}{\sqrt[4]{(x-1)^3 (x+2)^5}} \, dx$ is equal to (where $C$ is a constant of integration):

(A) $\dfrac{3}{4} \left( \dfrac{x+2}{x-1} \right)^{\frac{1}{4}} + C$

(B) $\dfrac{3}{4} \left( \dfrac{x+2}{x-1} \right)^{\frac{5}{4}} + C$

(D) $\dfrac{4}{3} \left( \dfrac{x-1}{x+2} \right)^{\frac{5}{4}} + C$

Click to View Answer

Correct Answer: [Insert Correct Option]

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Mathematics Questions

1. The integral $\int \frac{1}{\sqrt[4]{(x-1)^3 (x+2)^5}} \, dx$ is equal to (where $C$ is a constant of integration):

(A) $\dfrac{3}{4} \left( \dfrac{x+2}{x-1} \right)^{\frac{1}{4}} + C$

(B) $\dfrac{3}{4} \left( \dfrac{x+2}{x-1} \right)^{\frac{5}{4}} + C$

(D) $\dfrac{4}{3} \left( \dfrac{x-1}{x+2} \right)^{\frac{5}{4}} + C$

Click to View Answer

Correct Answer: [Insert Correct Option]

Mathematics Questions

1. The integral $\int \frac{1}{\sqrt[4]{(x-1)^3 (x+2)^5}} \, dx$ is equal to (where $C$ is a constant of integration):

(A) $\dfrac{3}{4} \left( \dfrac{x+2}{x-1} \right)^{\frac{1}{4}} + C$

(B) $\dfrac{3}{4} \left( \dfrac{x+2}{x-1} \right)^{\frac{5}{4}} + C$

(D) $\dfrac{4}{3} \left( \dfrac{x-1}{x+2} \right)^{\frac{5}{4}} + C$

Click to View Answer

Correct Answer: [Insert Correct Option]

Mathematics Questions

1. The integral $\int \frac{1}{\sqrt[4]{(x-1)^3 (x+2)^5}} \, dx$ is equal to (where $C$ is a constant of integration):

(A) $\dfrac{3}{4} \left( \dfrac{x+2}{x-1} \right)^{\frac{1}{4}} + C$

(B) $\dfrac{3}{4} \left( \dfrac{x+2}{x-1} \right)^{\frac{5}{4}} + C$

(D) $\dfrac{4}{3} \left( \dfrac{x-1}{x+2} \right)^{\frac{5}{4}} + C$

Click to View Answer

Correct Answer: [Let the integral be $I$ :

$I = \int \frac{1}{\sqrt[4]{(x-1)^3(x+2)^5}} \, dx$

First, rewrite the term inside the fourth root by multiplying and dividing by $(x-1)^5$ . This allows us to factor out a perfect fourth power:

$(x-1)^3(x+2)^5 = (x-1)^8 \cdot \frac{(x+2)^5}{(x-1)^5} = (x-1)^8 \left(\frac{x+2}{x-1}\right)^5$

Substitute this back into the integral:

$I = \int \frac{1}{\sqrt[4]{(x-1)^8 \left(\frac{x+2}{x-1}\right)^5}} \, dx = \int \frac{1}{(x-1)^2 \left(\frac{x+2}{x-1}\right)^{\frac{5}{4}}} \, dx$

Now, use the method of substitution. Let:

$t = \frac{x+2}{x-1}$

Differentiate $t$ with respect to $x$ using the quotient rule:

$dt = \frac{(1)(x-1) - (x+2)(1)}{(x-1)^2} \, dx$

$dt = \frac{x - 1 - x - 2}{(x-1)^2} \, dx = \frac{-3}{(x-1)^2} \, dx$

This allows us to isolate the $dx$ term:

$\frac{dx}{(x-1)^2} = -\frac{dt}{3}$

$t$ and $dx$ back into our integral $I$ :

$I = \int \frac{1}{t^{\frac{5}{4}}} \left(-\frac{dt}{3}\right) = -\frac{1}{3} \int t^{-\frac{5}{4}} \, dt$

Integrate using the standard power rule ( $\int x^n dx = \frac{x^{n+1}}{n+1}$ ): $I = -\frac{1}{3} \left( \frac{t^{-\frac{5}{4} + 1}}{-\frac{5}{4} + 1} \right) + C$

$I = -\frac{1}{3} \left( \frac{t^{-\frac{1}{4}}}{-\frac{1}{4}} \right) + C$

$I = \frac{4}{3} t^{-\frac{1}{4}} + C$

]

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EXAM GRID
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INDIA • AFTER CLASS 12 • 2026

Your Roadmap
After Class 12

Explore entrance exams, colleges, courses and career
pathways after Class 12 — all in one interactive roadmap.

12+
Major Pathways

20+
Entrance Exams

50+
College Options

∞
Career Possibilities

Complete Class 12

Your starting point

⚙️

PCM

Physics • Chemistry • Mathematics

🧬

PCB

Physics • Chemistry • Biology

📊

Commerce

Accounts • Economics • Business

🎨

Humanities

History • Political Science • Languages

🌐

Any Stream

Courses open across streams

Choose Your Entrance Path

Select an examination based on your desired career

ENGINEERING

JEE Main

Gateway to NITs, IIITs and many engineering
institutes. Also used for JEE Advanced eligibility.

PCM
B.Tech
National

Official website ↗

ENGINEERING

JEE Advanced

Entrance examination for undergraduate programmes
at the IITs.

PCM
IIT

Official website ↗

ENGINEERING

BITSAT

Admission route for undergraduate programmes
at BITS campuses.

PCM
BITS

Official website ↗

ENGINEERING

VITEEE

Engineering admission route for VIT.

PCM
B.Tech

Official website ↗

ENGINEERING

MET

Manipal Entrance Test for various undergraduate
programmes.

PCM
Engineering

Official website ↗

MEDICAL

NEET UG

Common national entrance route for undergraduate
medical education and specified AYUSH programmes.

PCB
MBBS
BDS

Official website ↗

UNIVERSITY

CUET UG

Common University Entrance Test for participating
universities and undergraduate programmes.

UG
Multiple Streams

Official website ↗

LAW

CLAT

Major national entrance route for undergraduate
law programmes at participating NLUs.

BA LLB
BBA LLB

Official website ↗

MANAGEMENT

IPMAT

Integrated management admission route at participating
IIMs.

BBA
MBA

IIM Indore ↗

DESIGN

NIFT Entrance

Entrance pathway for fashion and design-related
undergraduate programmes.

Design
Fashion

Official website ↗

ARCHITECTURE

NATA

National aptitude examination used by participating
architecture institutions for B.Arch admissions.

PCM
B.Arch

Official website ↗

DEFENCE

NDA / NA

UPSC entrance pathway for admission to the National
Defence Academy and Naval Academy.

Defence
UPSC

UPSC website ↗

HOSPITALITY

NCHM JEE

Entrance pathway for undergraduate hospitality
and hotel administration programmes.

Hospitality
B.Sc.

Official website ↗

AGRICULTURE

ICAR / CUET

Agriculture and allied undergraduate admission
pathways through participating institutions.

Agriculture
B.Sc.

ICAR website ↗

FINANCE

CA Foundation

Professional accounting pathway through the
Institute of Chartered Accountants of India.

Commerce
Finance

ICAI website ↗

FINANCE

CMA

Cost and management accounting professional pathway.

Commerce
Finance

ICMAI website ↗

CORPORATE

CS

Company Secretary professional qualification pathway.

Commerce
Corporate

ICSI website ↗

Engineering Pathway

PCM → Entrance → Institute → Engineering Career

⚙️ JEE / Engineering Route

Class 12 PCM
Physics + Chemistry + Mathematics

→

JEE Main
National Engineering Entrance

→

JEE Advanced
For IIT pathway

→

JoSAA
Seat allocation

→

Engineering College
B.Tech / B.E.

🏛️ Major Institute Options

IIT Bombay

IIT • Engineering

LEARNING

Mathematics Questions

Mathematics Questions

Mathematics Questions

Mathematics Questions

Mathematics Questions

Mathematics Questions

Your Roadmap After Class 12

Complete Class 12

PCM

PCB

Commerce

Humanities

Any Stream

Choose Your Entrance Path

JEE Main

JEE Advanced

BITSAT

VITEEE

MET

NEET UG

CUET UG

CLAT

IPMAT

NIFT Entrance

NATA

NDA / NA

NCHM JEE

ICAR / CUET

CA Foundation

CMA

CS

Engineering Pathway

⚙️ JEE / Engineering Route

🏛️ Major Institute Options

Medical & Healthcare Pathway

🧬 NEET Route

Commerce & Management

📊 Commerce Route

Law Pathway

⚖️ CLAT Route

Other Major Career Routes

Design

Architecture

Defence

Hotel Management

Agriculture

B.Sc. / Research

CA / CMA / CS

Journalism & Media

Quick Decision Map

Your Roadmap
After Class 12