JEE Mathematics Indefinite Integration MCQs Set 13

Answer: (a) \( - \frac{4 x^{\frac{5}{4}}}{5} + x + c \)

Explanation:
1. Since 1 - √x = (1 - x^(1/4))(1 + x^(1/4)), the integrand simplifies to 1 - x^(1/4).
2. Integral = x - (4/5)x^(5/4).
3. Differentiating this result returns the integrand, so option (a) is correct.

Teacher's Note:
1. Rationalise or simplify the integrand algebraically before integrating.
2. Substitution of a power of x often removes fractional exponents.

Answer: (a) \( - \frac{\cos{\left(5 x^{7} \right)}}{35} + c \)

Explanation:
1. Put 5x⁷ = t, so 35x⁶ dx = dt.
2. The integral is (1/35)∫sin t dt = -(1/35)cos t.
3. Writing this as (k/5)cos(5x⁷) gives k = -1/7.
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) \( e^{x} \sec{\left(x \right)} + c \)

Explanation:
1. For x near 0, √(1 + sin 2x) = sin x + cos x and 1 + cos 2x = 2cos²x.
2. The integrand is e^x (sec x tan x + sec x) = e^x(f + f') with f = sec x.
3. Result = e^x sec 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) \( \sqrt{2} e^{\frac{x}{2}} \sin{\left(\frac{x}{2} \right)} + c \)

Explanation:
1. Use ∫e^(ax) sin(bx + c) dx = e^(ax)[a sin(bx + c) - b cos(bx + c)]/(a² + b²) with a = b = 1/2.
2. The result is e^(x/2)[sin θ - cos θ] where θ = π/4 + x/2.
3. sin θ - cos θ = √2 sin(θ - π/4) = √2 sin(x/2).
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) \( - \csc{\left(x \right)} + \sec{\left(x \right)} + c \)

Explanation:
1. Split: sin³x/(sin²x cos²x) + cos³x/(sin²x cos²x) = sin x/cos²x + cos x/sin²x = ∫ sin x/cos²x dx = sec x and ∫ cos x/sin²x dx = -cosec x.
2. Result = sec x - cosec x.
3. 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: (c) \( \frac{2 \left(a + x\right)^{\frac{3}{2}} - 2 \left(b + x\right)^{\frac{3}{2}}}{3 a - 3 b} + c \)

Explanation:
1. Rationalise: multiply by (√(x+a) - √(x+b)) so the denominator becomes a - b.
2. The integrand is [√(x+a) - √(x+b)]/(a - b).
3. Integral = 2/(3(a - b)) [(x+a)^(3/2) - (x+b)^(3/2)].
4. Differentiating this result returns the integrand, so option (c) is correct.

Teacher's Note:
1. Rationalise or simplify the integrand algebraically before integrating.
2. Substitution of a power of x often removes fractional exponents.

Answer: (b) \( \frac{x^{5}}{5} - \frac{x^{4}}{4} + \frac{x^{3}}{3} - \frac{x^{2}}{2} + x + c \)

Explanation:
1. Since 1 + x⁵ = (1 + x)(1 - x + x² - x³ + x⁴), the integrand is 1 - x + x² - x³ + x⁴.
2. Integrating term by term gives x - x²/2 + x³/3 - x⁴/4 + x⁵/5.
3. Differentiating this result returns the integrand, so option (b) is correct.

Teacher's Note:
1. Rationalise or simplify the integrand algebraically before integrating.
2. Substitution of a power of x often removes fractional exponents.

Answer: (c) \( - 2 \sqrt{\cot{\left(x \right)}} + c \)

Explanation:
1. Write sin x √(sin x cos x) = sin²x √(cot x).
2. So the integrand is cosec²x/√(cot x). Put cot x = t.
3. Integral = -∫t^(-1/2) dt = -2√(cot 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: (a) \( \log{\left(\tan{\left(\frac{\tan{\left(x \right)}}{2} + \frac{\pi}{4} \right)} \right)} + c \)

Explanation:
1. Put tan x = t, so sec²x dx = dt.
2. The integral is ∫sec t dt = log|sec t + tan t| = log tan(π/4 + t/2).
3. Result = log tan(π/4 + (1/2)tan 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) \( \frac{x}{2} + \frac{\log{\left(\left|{\sin{\left(x \right)} - \cos{\left(x \right)}}\right| \right)}}{2} + c \)

Explanation:
1. Write sin x = (1/2)[(sin x - cos x) + (sin x + cos x)].
2. The integrand is 1/2 + (1/2)(cos x + sin x)/(sin x - cos x), where (sin x + cos x) is the derivative of (sin x - cos x).
3. Result = (1/2)[x + log|sin x - cos x|].
4. Differentiating this result returns the integrand, so option (c) 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) \( \frac{\log{\left(\left|{\sin{\left(3 x \right)}}\right| \right)}}{3} + c \)

Explanation:
1. Use tan x tan(60° - x) tan(60° + x) = tan 3x.
2. The integrand is cot 3x.
3. Integral = (1/3) log|sin 3x|.
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: (a) \( \frac{\sin^{-1}{\left(x^{3} \right)}}{3} + c \)

Explanation:
1. Put x³ = t, so x² dx = dt/3.
2. The integral becomes (1/3)∫dt/√(1 - t²) = (1/3) sin⁻¹t = (1/3) sin⁻¹(x³).
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: (b) \( \frac{4 \sqrt{\left(\sqrt{x} + 1\right)^{3}}}{3} + c \)

Explanation:
1. Put 1 + √x = t, so dx/√x = 2 dt.
2. The integral is 2∫t^(1/2) dt = (4/3)t^(3/2).
3. Result = (4/3)(1 + √x)^(3/2).
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) \( \left(1 - \frac{1}{x}\right) \log{\left(1 - x \right)} - \log{\left(x \right)} + c \)

Explanation:
1. By parts with u = log(1 - x) and v' = 1/x²: ∫ = -log(1 - x)/x - ∫dx/(x(1 - x)).
2. ∫dx/(x(1 - x)) = log x - log(1 - x).
3. Result = (1 - 1/x) log(1 - x) - log x.
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: (c) \( x^{x} e^{x} + c \)

Explanation:
1. d/dx(x^x) = x^x(1 + log x).
2. Take f = x^x: f + f' = x^x(2 + log x).
3. Result = e^x x^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: (b) \( \frac{\sqrt{6} \tan^{-1}{\left(\frac{\sqrt{6} \tan{\left(\frac{x}{2} \right)}}{6} \right)}}{6} + c \)

Explanation:
1. Put tan(x/2) = t, cos x = (1 - t²)/(1 + t²) and dx = 2dt/(1 + t²).
2. 7 + 5cos x = (12 + 2t²)/(1 + t²), so the integral is ∫dt/(t² + 6).
3. Result = (1/√6) tan⁻¹(tan(x/2)/√6).
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) \( \frac{x^{2 x}}{2} + c \)

Explanation:
1. d/dx(x^x) = x^x(1 + log x).
2. Put x^x = t: the integral is ∫t dt = t²/2.
3. Result = (x^x)²/2.
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: (d) \( \frac{x^{2} e^{2 x}}{2} + c \)

Explanation:
1. Put x e^x = t, so (1 + x)e^x dx = dt.
2. The integrand is (x e^x)(e^x(1 + x)) dx = t dt.
3. Integral = t²/2 = (x e^x)²/2.
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) \( \frac{2 \sin^{-1}{\left(\left(\frac{x}{a}\right)^{\frac{3}{2}} \right)}}{3} + c \)

Explanation:
1. Put x^(3/2) = t, so (3/2)x^(1/2) dx = dt.
2. The integral is (2/3)∫dt/√(a³ - t²) = (2/3) sin⁻¹(t/a^(3/2)).
3. Result = (2/3) sin⁻¹((x/a)^(3/2)).
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: (a) \( - \frac{\cos{\left(x \right)}}{a \sin{\left(x \right)} + b} + c \)

Explanation:
1. Differentiate cos x/(b + a sin x): the derivative is [-sin x(b + a sin x) - a cos²x]/(b + a sin x)² = -(a + b sin x)/(b + a sin x)².
2. Hence the integral of (a + b sin x)/(b + a sin x)² is -cos x/(b + a sin x).
3. 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: (b) \( - e^{x} \cot{\left(\frac{x}{2} \right)} + c \)

Explanation:
1. Write (1 - sin x)/(1 - cos x) = (1/2)cosec²(x/2) - cot(x/2).
2. With f = -cot(x/2), f' = (1/2)cosec²(x/2), so the integrand is e^x(f + f').
3. Result = -e^x cot(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: (b) \( \frac{\sqrt{3} \log{\left(\left|{\frac{\sqrt{3} \tan{\left(x \right)} + 1}{\sqrt{3} \tan{\left(x \right)} - 1}}\right| \right)}}{6} + c \)

Explanation:
1. cos 3x = cos x(4cos²x - 3), so the integrand is 1/(4cos²x - 3).
2. Divide by cos²x: ∫sec²x dx/(4 - 3 sec²x) = ∫sec²x dx/(1 - 3tan²x).
3. Put tan x = t: ∫dt/(1 - 3t²) = (1/(2√3)) log|(1 + √3t)/(1 - √3t)|.
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.