NCERT Class 12 · Ch 7 Exercise 7.2 · Q5
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.2 Question 5
Evaluate \(\int \sin{\left(a x + b \right)} \cos{\left(a x + b \right)}\, dx\).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: For a≠0: sin²(ax+b)/(2a)+C. For a=0: x sin b cos b+C.
- Work on each open interval where the original integrand is real and defined.
- u=sin(ax+b), du=a cos(ax+b)dx, a≠0. If a=0, the answer is x sin b cos b+C.
- An antiderivative is F(x)=\(\frac{\sin^{2}{\left(a x + b \right)}}{2 a}\).
- Differentiate this expression to check that F′(x)=\(\sin{\left(a x + b \right)} \cos{\left(a x + b \right)}\). Add an arbitrary constant C on each domain component.
Master Chapter 7: Integrals
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
← Previous Question
Evaluate \(\int \sin{\left(x \right)} \sin{\left(\cos{\left(x…
Next Question →Evaluate \(\int \sqrt{a x + b}\, dx\).
All Questions in Integrals — Exercise 7.2
Related Questions in Integrals
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int x^2e^{x^3}\,dx\).
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int e^x(1+x)\sec^2(xe^x)\,dx\) on an interval where defined.
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int e^x(\cos x-\sin x)\,dx\).
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int dx/(\sin^2x\cos^2x)\) on an interval avoiding its poles.
Class 12 Integrals · Mcq Single Correct
If \(\int\frac{3e^x-5e^{-x}}{4e^x+5e^{-x}}dx=ax+b\ln|4e^x+5e^{-x}|+C\), find a,b.
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).