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NCERT Class 12 Maths Chapter 7: Integrals 40 Questions Solved

NCERT Solutions for Class 12 Maths Chapter 7 Integrals — Miscellaneous Exercise

Click any question below for its step-by-step NCERT textbook solution, concept hints, and worked explanation.

Evaluate \(\int \frac{1}{- x^{3} + x}\, dx\).

Evaluate \(\int \frac{1}{\sqrt{a + x} + \sqrt{b + x}}\, dx\).

Evaluate \(\int \frac{1}{x \sqrt{a x - x^{2}}}\, dx\).

Evaluate \(\int \frac{1}{x^{2} \left(x^{4} + 1\right)^{\frac{3}{4}}}\, dx\).

Evaluate \(\int \frac{1}{\sqrt[3]{x} + \sqrt{x}}\, dx\).

Evaluate \(\int \frac{5 x}{\left(x + 1\right) \left(x^{2} + 9\right)}\, dx\).

Evaluate \(\int \left(- \frac{\sin{\left(x \right)}}{\sin{\left(a - x \right)}}\right)\, dx\).

Evaluate \(\int \frac{x^{5} - x^{4}}{x^{3} - x^{2}}\, dx\). Original logarithm domain: x>0, x≠1.

Evaluate \(\int \frac{\cos{\left(x \right)}}{\sqrt{4 - \sin^{2}{\left(x \right)}}}\, dx\).

Evaluate \(\int \frac{\sin^{8}{\left(x \right)} - \cos^{8}{\left(x \right)}}{- 2 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}…

Evaluate \(\int \frac{1}{\cos{\left(a + x \right)} \cos{\left(b + x \right)}}\, dx\).

Evaluate \(\int \frac{x^{3}}{\sqrt{1 - x^{8}}}\, dx\).

Evaluate \(\int \frac{e^{x}}{\left(e^{x} + 1\right) \left(e^{x} + 2\right)}\, dx\).

Evaluate \(\int \frac{1}{\left(x^{2} + 1\right) \left(x^{2} + 4\right)}\, dx\).

Evaluate \(\int \sin{\left(x \right)} \cos^{3}{\left(x \right)}\, dx\). Original logarithm domain: sin x>0.

Evaluate \(\int \frac{x^{3}}{x^{4} + 1}\, dx\). Original logarithm domain: x>0.

Integrate f′(ax+b)[f(ax+b)]ⁿ.

Evaluate \(\int \frac{1}{\sqrt{\sin^{3}{\left(x \right)} \sin{\left(\alpha + x \right)}}}\, dx\).

Evaluate \(\int \sqrt{\frac{1 - \sqrt{x}}{\sqrt{x} + 1}}\, dx\).

Evaluate \(\int \frac{\left(\sin{\left(2 x \right)} + 2\right) e^{x}}{\cos{\left(2 x \right)} + 1}\, dx\).

Evaluate \(\int \frac{x^{2} + x + 1}{\left(x + 1\right)^{2} \left(x + 2\right)}\, dx\).

Evaluate \(\int \operatorname{atan}{\left(\sqrt{\frac{1 - x}{x + 1}} \right)}\, dx\).

Evaluate \(\int \frac{\sqrt{x^{2} + 1} \left(- 2 \log{\left(x \right)} + \log{\left(x^{2} + 1 \right)}\right)}{x^{4}}\, dx\).

Evaluate \(\int\limits_{\frac{\pi}{2}}^{\pi} \frac{\left(1 - \sin{\left(x \right)}\right) e^{x}}{1 - \cos{\left(x \right)}}\, dx\).

Evaluate \(\int\limits_{0}^{\frac{\pi}{4}} \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sin^{4}{\left(x \right)} +…

Evaluate \(\int\limits_{0}^{\frac{\pi}{2}} \frac{\cos^{2}{\left(x \right)}}{4 \sin^{2}{\left(x \right)} + \cos^{2}{\left(x…

Evaluate \(\int\limits_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin{\left(x \right)} + \cos{\left(x \right)}}{\sqrt{\sin{\left(2 x…

Evaluate \(\int\limits_{0}^{1} \frac{1}{- \sqrt{x} + \sqrt{x + 1}}\, dx\).

Evaluate \(\int\limits_{0}^{\frac{\pi}{4}} \frac{\sin{\left(x \right)} + \cos{\left(x \right)}}{16 \sin{\left(2 x \right)} + 9}\, dx\).

Evaluate \(\int\limits_{0}^{\frac{\pi}{2}} \sin{\left(2 x \right)} \operatorname{atan}{\left(\sin{\left(x \right)} \right)}\, dx\).

Evaluate ∫₁⁴[|x−1|+|x−2|+|x−3|]dx.

Evaluate \(\int\limits_{1}^{3} \frac{1}{x^{2} \left(x + 1\right)}\, dx\).

Evaluate \(\int\limits_{0}^{1} x e^{x}\, dx\).

Prove ∫₋₁¹x¹⁷cos⁴x dx=0.

Evaluate \(\int\limits_{0}^{\frac{\pi}{2}} \sin^{3}{\left(x \right)}\, dx\).

Evaluate \(\int\limits_{0}^{\frac{\pi}{4}} 2 \tan^{3}{\left(x \right)}\, dx\).

Evaluate \(\int\limits_{0}^{1} \operatorname{asin}{\left(x \right)}\, dx\).

Evaluate \(\int \frac{1}{e^{x} + e^{- x}}\, dx\).

Evaluate \(\int \frac{\cos{\left(2 x \right)}}{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}\, dx\).

If f(a+b−x)=f(x), choose the expression for ∫ₐᵇx f(x)dx.