JEE Mathematics Indefinite Integration MCQs Set 15

Answer: (b) \( \frac{\log{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}^{2}}{2} + c \)

Explanation:
1. Put log(sec x + tan x) = t; then dt = sec x dx.
2. The integral is ∫t dt = t²/2.
3. Result = (1/2)[log(sec x + tan x)]².
4. Differentiating this result returns the integrand, so option (b) is correct.

Teacher's Note:
1. Choose the substitution so that the derivative of the inner function appears in the integrand.
2. After substituting, change every part of the integral to the new variable.

Answer: (b) \( e^{\frac{x - 1}{x}} + c \)

Explanation:
1. (x - 1)/x = 1 - 1/x, and d(1 - 1/x) = dx/x².
2. So the integral is ∫e^t dt with t = 1 - 1/x.
3. Result = e^((x-1)/x).
4. Differentiating this result returns the integrand, so option (b) is correct.

Teacher's Note:
1. Choose the substitution so that the derivative of the inner function appears in the integrand.
2. After substituting, change every part of the integral to the new variable.

Answer: (d) \( x + c \)

Explanation:
1. cosh x + sinh x = e^x, so the integrand is e^x/e^x = 1.
2. Integral = x.
3. Differentiating this result returns the integrand, so option (d) is correct.

Teacher's Note:
1. Learn the standard integrals thoroughly; most one-line questions reduce to them after a small rewrite.
2. Always verify by differentiating the answer when options look similar.

Answer: (c) \( \log{\left(\left|{\sin{\left(x \right)} + \cos{\left(x \right)}}\right| \right)} + c \)

Explanation:
1. (1 - tan x)/(1 + tan x) = (cos x - sin x)/(cos x + sin x).
2. The numerator is the derivative of the denominator.
3. Result = log|sin x + cos x|.
4. Differentiating this result returns the integrand, so option (c) is correct.

Teacher's Note:
1. Learn the standard integrals thoroughly; most one-line questions reduce to them after a small rewrite.
2. Always verify by differentiating the answer when options look similar.

Answer: (d) \( \frac16\tan^6x-\frac14\tan^4x+\frac12\tan^2x \)

Explanation:
1. With I_n = ∫tanⁿx dx = tan^(n-1)x/(n - 1) - I_(n-2).
2. I₇ = tan⁶x/6 - I₅ = tan⁶x/6 - tan⁴x/4 + I₃ = tan⁶x/6 - tan⁴x/4 + tan²x/2 - I₁.
3. I₁ = -log|cos x| = log|sec x|, so -I₁ = log|cos x|.
4. Differentiating this result returns the integrand, so option (d) is correct.

Teacher's Note:
1. Convert products and powers of trigonometric functions to sums using identities, or substitute for the simplest function.
2. Check the final form against the options by differentiation.

Answer: (c) \( \log{\left(e^{x} \left|{\frac{x}{x e^{x} + 1}}\right| \right)} + c \)

Explanation:
1. Multiply numerator and denominator by e^x: put x e^x = t, so (x + 1)e^x dx = dt.
2. The integral is ∫dt/(t(1 + t)) = log|t/(1 + t)|.
3. Result = log|x e^x/(1 + x e^x)|.
4. Differentiating this result returns the integrand, so option (c) is correct.

Teacher's Note:
1. Choose the substitution so that the derivative of the inner function appears in the integrand.
2. After substituting, change every part of the integral to the new variable.

Answer: (b) \( x \left(\log{\left(x \right)}^{2} - 9 \log{\left(x \right)} + 18\right) + c \)

Explanation:
1. ∫(log x)² dx = x[(log x)² - 2 log x + 2] and ∫log x dx = x(log x - 1).
2. Total = x[L² - 2L + 2 - 7L + 7 + 9] with L = log x.
3. Result = x[(log x)² - 9 log x + 18].
4. Differentiating this result returns the integrand, so option (b) is correct.

Teacher's Note:
1. Use integration by parts with the ILATE order for choosing u.
2. Special forms such as the integral of e to the x times (f + f') are solved at once.

Answer: (a) \( f(x)g'(x)-g(x)f'(x)+c \)

Explanation:
1. d/dx[f g' - g f'] = f' g' + f g'' - g' f' - g f'' = f g'' - f'' g.
2. So the integrand is the derivative of f g' - g f'.
3. Result = f(x)g'(x) - g(x)f'(x) + c.
4. Differentiating this result returns the integrand, so option (a) is correct.

Teacher's Note:
1. Use the definition that the integral is the function whose derivative is the integrand.
2. Differentiating the options is the fastest check.

Answer: (a) \( - 2 \sqrt{x} \cos{\left(\sqrt{x} \right)} + 2 \sin{\left(\sqrt{x} \right)} + c \)

Explanation:
1. Put √x = t, so dx = 2t dt.
2. The integral is 2∫t sin t dt = 2(sin t - t cos t).
3. Result = 2(sin √x - √x cos √x).
4. Differentiating this result returns the integrand, so option (a) is correct.

Teacher's Note:
1. Use integration by parts with the ILATE order for choosing u.
2. Special forms such as the integral of e to the x times (f + f') are solved at once.

Answer: (a) \( \log{\left(\left|{\frac{x}{x + \cos{\left(x \right)}}}\right| \right)} + c \)

Explanation:
1. d/dx[(x + cos x)/x] = -(cos x + x sin x)/x².
2. Hence d/dx log[(x + cos x)/x] = -(cos x + x sin x)/(x(x + cos x)).
3. Integral = -log|(x + cos x)/x| = log|x/(x + cos x)|.
4. Differentiating this result returns the integrand, so option (a) is correct.

Teacher's Note:
1. Choose the substitution so that the derivative of the inner function appears in the integrand.
2. After substituting, change every part of the integral to the new variable.

Answer: (a) \( - \frac{\cot^{n}{\left(x \right)}}{n} + c \)

Explanation:
1. Write the integrand as cot^(n-1)x · cosec²x.
2. Put cot x = t, so cosec²x dx = -dt.
3. Integral = -∫t^(n-1) dt = -cotⁿx/n.
4. Differentiating this result returns the integrand, so option (a) is correct.

Teacher's Note:
1. Choose the substitution so that the derivative of the inner function appears in the integrand.
2. After substituting, change every part of the integral to the new variable.

Answer: (a) \( \log{\left(\left|{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}\right| \right)} + c \)

Explanation:
1. d/dx[log tan(x/2)] = (1/2)sec²(x/2)/tan(x/2) = 1/sin x = cosec x.
2. So the integral has the form ∫dt/t with t = log tan(x/2).
3. Result = log|log tan(x/2)|.
4. Differentiating this result returns the integrand, so option (a) is correct.

Teacher's Note:
1. Choose the substitution so that the derivative of the inner function appears in the integrand.
2. After substituting, change every part of the integral to the new variable.

Answer: (a) \( \frac{\left(x - 1\right) e^{x}}{x + 1} + c \)

Explanation:
1. Write (x² + 1)/(x + 1)² = (x - 1)/(x + 1) + 2/(x + 1)².
2. With f = (x - 1)/(x + 1), f' = 2/(x + 1)², so the integrand is e^x(f + f').
3. Result = e^x (x - 1)/(x + 1).
4. Differentiating this result returns the integrand, so option (a) is correct.

Teacher's Note:
1. Use integration by parts with the ILATE order for choosing u.
2. Special forms such as the integral of e to the x times (f + f') are solved at once.

Answer: (d) \( \frac2n\sin nx \)

Explanation:
1. sin(n + 1)x - sin(n - 1)x = 2 cos nx sin x.
2. So (sin(n + 1)x - sin(n - 1)x)/sin x = 2 cos nx.
3. I_(n+1) - I_(n-1) = ∫2 cos nx dx = (2/n) sin nx.
4. Differentiating this result returns the integrand, so option (d) is correct.

Teacher's Note:
1. Convert products and powers of trigonometric functions to sums using identities, or substitute for the simplest function.
2. Check the final form against the options by differentiation.

Answer: (b) \( \frac{\sin^{-1}{\left(2^{x} \right)}}{\log{\left(2 \right)}} + c \)

Explanation:
1. Put 2^x = t, so 2^x log 2 dx = dt.
2. The integral is (1/log 2)∫dt/√(1 - t²) = (1/log 2) sin⁻¹t.
3. So k = 1/log 2.
4. Differentiating this result returns the integrand, so option (b) is correct.

Teacher's Note:
1. Choose the substitution so that the derivative of the inner function appears in the integrand.
2. After substituting, change every part of the integral to the new variable.

Answer: (b) \( \frac{\sec^{3}{\left(2 x \right)}}{6} - \frac{\sec{\left(2 x \right)}}{2} + c \)

Explanation:
1. Write tan³2x sec 2x = (sec²2x - 1) sec 2x tan 2x and put sec 2x = t, dt = 2 sec 2x tan 2x dx.
2. Integral = (1/2)∫(t² - 1) dt = (1/6)(t³ - 3t).
3. Result = (1/6)(sec³2x - 3 sec 2x).
4. Differentiating this result returns the integrand, so option (b) is correct.

Teacher's Note:
1. Choose the substitution so that the derivative of the inner function appears in the integrand.
2. After substituting, change every part of the integral to the new variable.

Answer: (a) \( \frac{\left(x + 2\right) \sqrt{2 x + 1}}{3} + c \)

Explanation:
1. Put √(2x + 1) = t, so x = (t² - 1)/2, x + 1 = (t² + 1)/2 and dx = t dt.
2. The integral is (1/2)∫(t² + 1) dt = t³/6 + t/2 = t(t² + 3)/6.
3. Result = √(2x + 1)(x + 2)/3.
4. Differentiating this result returns the integrand, so option (a) is correct.

Teacher's Note:
1. Choose the substitution so that the derivative of the inner function appears in the integrand.
2. After substituting, change every part of the integral to the new variable.

Answer: (b) \( \frac{\left(x \log{\left(x \right)} - x\right)^{2}}{2} + c \)

Explanation:
1. d/dx(x log x - x) = log x.
2. The integrand is x(log x - 1) log x = (x log x - x) · d/dx(x log x - x).
3. Integral = (1/2)(x log x - x)².
4. Differentiating this result returns the integrand, so option (b) is correct.

Teacher's Note:
1. Choose the substitution so that the derivative of the inner function appears in the integrand.
2. After substituting, change every part of the integral to the new variable.

Answer: (c) \( - \frac{x^{4} \cot{\left(2 \log{\left(x \right)} \right)}}{4} + c \)

Explanation:
1. Put log x = t and 4t = u: the integral is (1/4)∫e^u (1 - sin u)/(1 - cos u) du.
2. (1 - sin u)/(1 - cos u) = (1/2)cosec²(u/2) - cot(u/2) = f + f' with f = -cot(u/2).
3. Result = -(1/4)e^u cot(u/2) = -(x⁴/4) cot(2 log x).
4. Differentiating this result returns the integrand, so option (c) is correct.

Teacher's Note:
1. Use integration by parts with the ILATE order for choosing u.
2. Special forms such as the integral of e to the x times (f + f') are solved at once.

Answer: (a) \( e^{x \sec{\left(x \right)}} + c \)

Explanation:
1. Let t = x sec x; dt = (sec x + x sec x tan x) dx = sec x(1 + x tan x) dx.
2. The integral is ∫e^t dt = e^t = e^(x sec x).
3. Differentiating this result returns the integrand, so option (a) is correct.

Teacher's Note:
1. Choose the substitution so that the derivative of the inner function appears in the integrand.
2. After substituting, change every part of the integral to the new variable.

Answer: (c) \( \frac{\sqrt{2} \tan^{-1}{\left(\frac{\sqrt{2} \left(x^{2} - 1\right)}{2 x} \right)}}{2} + c \)

Explanation:
1. Divide numerator and denominator by x²: (1 + 1/x²)/(x² + 1/x²).
2. Put x - 1/x = t, so (1 + 1/x²) dx = dt and x² + 1/x² = t² + 2.
3. Integral = ∫dt/(t² + 2) = (1/√2) tan⁻¹((x² - 1)/(√2 x)).
4. Differentiating this result returns the integrand, so option (c) is correct.

Teacher's Note:
1. Choose the substitution so that the derivative of the inner function appears in the integrand.
2. After substituting, change every part of the integral to the new variable.