JEE Mathematics Indefinite Integration MCQs Set 17

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

Explanation:
1. 2 + 2 sin 2x = 2(cos x + sin x)².
2. The integrand is 2(cos x - sin x)(cos x + sin x) = 2 cos 2x.
3. Integral = sin 2x.
4. Differentiating this result returns the integrand, so option (a) 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) \( x + \log{\left(\left|{\cos{\left(x + \frac{\pi}{4} \right)}}\right| \right)} + c \)

Explanation:
1. Write sin x = sin[(x - π/4) + π/4] = (1/√2)[sin(x - π/4) + cos(x - π/4)].
2. The integrand is 1 + cot(x - π/4).
3. Integral = x + log|sin(x - π/4)|.
4. Differentiating this result returns the integrand, so option (b) 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) \( \sqrt{x^{2} + 1} + c \)

Explanation:
1. If θ = tan⁻¹x, then sin θ = x/√(1 + x²).
2. Integral = ∫x/√(1 + x²) dx = √(1 + x²).
3. 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: (c) \( \frac{x \log{\left(\frac{x}{e} \right)}}{\log{\left(10 \right)}} + c \)

Explanation:
1. log₁₀x = log x · log₁₀e.
2. ∫log x dx = x log x - x = x log(x/e).
3. Result = log₁₀e · x log_e(x/e).
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: (d) \( \frac{x}{2} + \frac{\tan{\left(x \right)}}{2} + c \)

Explanation:
1. 1 + cos 2x = 2cos²x, so the integrand is (1 + cos²x)/(2cos²x) = (1/2)sec²x + 1/2.
2. Integral = (1/2)(tan x + x).
3. 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) \( - 2 x \sin{\left(\alpha \right)} + 2 \cos{\left(x \right)} + c \)

Explanation:
1. cos 2x - cos 2α = 2(sin²α - sin²x) = 2(sin α - sin x)(sin α + sin x).
2. So the integrand is -2(sin x + sin α).
3. Integral = 2cos x - 2x sin α = 2[cos x - x sin α].
4. Differentiating this result returns the integrand, so option (b) 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) \( - \frac{1}{2 \left(\tan{\left(x \right)} + \sec{\left(x \right)}\right)^{2}} + c \)

Explanation:
1. Put sec x + tan x = t, so sec x(sec x + tan x) dx = dt, i.e. sec x dx = dt/t.
2. The integral is ∫dt/t³ = -1/(2t²).
3. Result = -1/[2(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) \( 2 e^{\frac{x}{2}} \sec{\left(2 x \right)} + c \)

Explanation:
1. Write the integrand as e^(x/2)[sec 2x + 4 sec 2x tan 2x].
2. With f = 2 sec 2x, we get f' = 4 sec 2x tan 2x, and e^(x/2) behaves like e^(ax) with a = 1/2.
3. Check: d/dx[2e^(x/2) sec 2x] = e^(x/2) sec 2x + 4e^(x/2) sec 2x tan 2x.
4. Result = 2e^(x/2) sec 2x.
5. 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: (b) \( x \tan{\left(\frac{\log{\left(x \right)}}{2} \right)} + c \)

Explanation:
1. Put log x = t, x = e^t: ∫e^t (1 + sin t)/(1 + cos t) dt.
2. (1 + sin t)/(1 + cos t) = (1/2)sec²(t/2) + tan(t/2), which is f' + f for f = tan(t/2).
3. Result = e^t tan(t/2) = x tan((log x)/2).
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: (d) \( 2 e^{\tan^{-1}{\left(\sqrt{x} \right)}} + c \)

Explanation:
1. The denominator is √x(1 + x).
2. Put tan⁻¹√x = t, so dt = dx/(2√x(1 + x)).
3. The integral is 2∫e^t dt = 2e^t.
4. Differentiating this result returns the integrand, so option (d) 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) \( \frac{x}{\sqrt{\log{\left(x \right)}^{2} + 1}} + c \)

Explanation:
1. Put log x = t, so the integral is ∫e^t (t² - t + 1)/(t² + 1)^(3/2) dt.
2. Take f = 1/√(t² + 1); then f' = -t/(t² + 1)^(3/2), and f + f' = (t² + 1 - t)/(t² + 1)^(3/2).
3. The integral is e^t f = x/√((log x)² + 1).
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) \( \log{\left(10^{x} + x^{10} \right)} + c \)

Explanation:
1. The numerator is 10^x log 10 + 10x⁹, which is exactly the derivative of the denominator 10^x + x¹⁰.
2. Hence the integral is log|10^x + x¹⁰|.
3. Differentiating this result returns the integrand, so option (a) 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: (b) \( 1,\ 1 \)

Explanation:
1. Write 9 cos x - sin x = A(4 cos x - 5 sin x) + B(4 sin x + 5 cos x).
2. Comparing coefficients: 4A + 5B = 9 and -5A + 4B = -1, so A = 1, B = 1.
3. The integral is log|4 sin x + 5 cos x| + x.
4. Differentiating this result returns the integrand, so option (b) 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: (a) \( 2 e^{\frac{x}{2}} + 2 \log{\left(\left|{e^{\frac{x}{2}} - 1}\right| \right)} - 2 + c \)

Explanation:
1. Put e^(x/2) = t, so e^x dx = 2t dt.
2. The integral becomes 2t/(t - 1) dt = 2[1 + 1/(t - 1)] dt = 2[t + log|t - 1|].
3. Back-substituting gives the result.
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: (c) \( x+\sqrt{1+x^2} \)

Explanation:
1. 1/(√(1 + x²) - x) = √(1 + x²) + x. Put z = x + √(1 + x²).
2. Then x = (z - 1/z)/2 and dx = (1/2)(1 + 1/z²) dz.
3. The integral is (1/2)∫zⁿ(1 + z⁻²) dz = (1/2)[z^(n+1)/(n + 1) + z^(n-1)/(n - 1)].
4. So z = x + √(1 + x²).
5. 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) \( - \frac{2 \sqrt{\frac{- q + x}{- p + x}}}{p - q} + c \)

Explanation:
1. The integrand is (x - p)^(-3/2)(x - q)^(-1/2) = (x - p)⁻²·√((x - p)/(x - q)) = (x - p)⁻² t^(-1/2) with t = (x - q)/(x - p).
2. dt = (q - p) dx/(x - p)², so the integral is (1/(q - p))∫t^(-1/2) dt = 2√t/(q - p).
3. Result = -(2/(p - q))√((x - q)/(x - p)).
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) \( \sin^{-1}{\left(\frac{e^{x}}{3} + \frac{2}{3} \right)} + c \)

Explanation:
1. Put e^x = t: the integral is ∫dt/√(5 - 4t - t²).
2. 5 - 4t - t² = 9 - (t + 2)².
3. Integral = sin⁻¹((t + 2)/3) = sin⁻¹((e^x + 2)/3).
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: (c) \( xf^{-1}(x)-g\left[f^{-1}(x)\right]+c \)

Explanation:
1. Put x = f(t), so f⁻¹(x) = t and dx = f'(t) dt.
2. ∫f⁻¹(x) dx = ∫t f'(t) dt = t f(t) - ∫f(t) dt = t f(t) - g(t).
3. Replace t by f⁻¹(x): x f⁻¹(x) - g(f⁻¹(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) \( \frac{\sec^{n}{\left(x \right)}}{n} + c \)

Explanation:
1. Write sec^n x tan x = sec^(n-1)x · (sec x tan x).
2. Put sec x = t, so sec x tan x dx = dt.
3. Integral = ∫t^(n-1) dt = sec^n x/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) \( \frac{2 \sqrt{x} - 2}{\sqrt{1 - x}} + c \)

Explanation:
1. Put √x = sin θ, so x = sin²θ and dx = 2 sin θ cos θ dθ; √(x - x²) = sin θ cos θ.
2. The integral is ∫2 dθ/(1 + sin θ) = 2(tan θ - sec θ).
3. tan θ - sec θ = (sin θ - 1)/cos θ = (√x - 1)/√(1 - x), so the result is 2(√x - 1)/√(1 - 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: (c) \( - \frac{5 \left(-1 + \frac{1}{x^{4}}\right)^{\frac{6}{5}}}{24} + c \)

Explanation:
1. Write (x - x⁵)^(1/5) = x(1/x⁴ - 1)^(1/5).
2. The integrand is (1/x⁴ - 1)^(1/5)/x⁵. Put 1/x⁴ - 1 = t, so dx/x⁵ = -dt/4.
3. Integral = -(1/4)(5/6)t^(6/5) = -(5/24)(1/x⁴ - 1)^(6/5).
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.