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Day-102 | Ex-7.3 Integration using Trigonometric Identities | Class-12 NCERT Solutions | Ashu Sir JAC 2026

📖 Description:
In this video (Day-102), Ashu Sir explains Integration using Trigonometric Identities from Exercise 7.3 Class-12 NCERT Maths step by step.
This method is one of the most important techniques for solving integration problems in Class-12 Board Exams as well as for JEE & other competitive exams.

✨ Topics Covered:

Basics of trigonometric identities in integration

Standard formulas and shortcuts

NCERT Ex-7.3 important questions solved in detail

Board exam & JEE Main level approach


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Solved examples — Integration using trigonometric identities


---

Example 1 —

Write . Let , .

\begin{aligned}
\int \sin^{3}x\,dx &= \int \sin x(1-\cos^{2}x)\,dx
= -\int (1-u^{2})\,du\\
&= -\Big(u-\frac{u^{3}}{3}\Big)+C
= -\cos x+\frac{\cos^{3}x}{3}+C.
\end{aligned}


---

Example 2 —

Use the double-angle identity .

\int \sin^{2}x\,dx = \int \frac{1-\cos2x}{2}\,dx
= \frac{x}{2}-\frac{\sin2x}{4}+C.


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Example 3 —

Express as . Put , ; note .

\begin{aligned}
\int \sin^{2}x\cos^{3}x\,dx
&= \int \sin^{2}x(1-\sin^{2}x)\cos x\,dx\\
&= \int (u^{2}-u^{4})\,du
= \frac{u^{3}}{3}-\frac{u^{5}}{5}+C\\
&= \frac{\sin^{3}x}{3}-\frac{\sin^{5}x}{5}+C.
\end{aligned}


---

Example 4 —

Use and then .

\begin{aligned}
\int \sin^{2}x\cos^{2}x\,dx
&= \int \frac{1}{4}\sin^{2}2x\,dx
= \frac{1}{4}\int \frac{1-\cos4x}{2}\,dx\\
&= \frac{1}{8}x-\frac{1}{32}\sin4x + C.
\end{aligned}


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Example 5 — (standard result)

Use integration by parts: write and take .

\begin{aligned}
\int \sec^{3}x\,dx &= \sec x\tan x-\int \tan x\cdot \sec x\tan x\,dx\\
&= \sec x\tan x-\int \sec x\tan^{2}x\,dx\\
&= \sec x\tan x-\int \sec x(\sec^{2}x-1)\,dx\\
&= \sec x\tan x-\int \sec^{3}x\,dx+\int \sec x\,dx.
\end{aligned}

2\int \sec^{3}x\,dx=\sec x\tan x+\int \sec x\,dx.

\boxed{\displaystyle \int \sec^{3}x\,dx=\tfrac{1}{2}\sec x\tan x+\tfrac{1}{2}\ln\big|\sec x+\tan x\big|+C.}


---

Example 6 —

Use .

\int \tan^{2}x\,dx=\int(\sec^{2}x-1)\,dx=\tan x - x + C.


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If you want, I can:

Turn these into a one-page PDF (ready to attach) and give you a download link, or

Produce 8–10 more Ex-7.3 solved questions (with progressively harder ones) for students, or


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