JEE Mathematics Indefinite Integration MCQs Set 16

Answer: (b) \( f(x)=\sin^{-1}x,\ g(x)=\frac{x-a}{a} \)

Explanation:
1. Complete the square: 2ax - x² = a² - (x - a)².
2. ∫dx/√(a² - (x - a)²) = sin⁻¹((x - a)/a) + c.
3. So f = sin⁻¹ and g(x) = (x - a)/a.
4. Differentiating this result returns the integrand, so option (b) 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) \( x \)

Explanation:
1. ∫cot⁴x dx = ∫cot²x(cosec²x - 1) dx = -cot³x/3 - ∫(cosec²x - 1) dx = -cot³x/3 + cot x + x + C.
2. So f(x) = x + C.
3. f(π/2) = π/2 gives C = 0, so f(x) = 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) \( \frac{\tan^{-1}{\left(\frac{3 \tan{\left(x \right)}}{4} \right)}}{12} + c \)

Explanation:
1. Divide numerator and denominator by cos²x: ∫sec²x dx/(9 tan²x + 16).
2. Put tan x = t: ∫dt/(9t² + 16) = (1/12) tan⁻¹(3t/4).
3. Result = (1/12) tan⁻¹(3 tan x/4).
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) \( \frac{\sec^{-1}{\left(\frac{a x}{b} \right)}}{b} + c \)

Explanation:
1. Put ax = t: the integral becomes ∫ dt/(t√(t² - b²)).
2. The standard form gives (1/b) sec⁻¹(t/b) = (1/b) sec⁻¹(ax/b).
3. 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: (a) \( \sin^{-1}{\left(\frac{\sin{\left(x \right)}}{3} + \frac{\cos{\left(x \right)}}{3} \right)} + c \)

Explanation:
1. Put sin x + cos x = t, so (cos x - sin x) dx = dt and sin 2x = t² - 1.
2. 8 - sin 2x = 9 - t².
3. Integral = sin⁻¹(t/3), so a = 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) \( \log{\left(\left|{x}\right| \right)} + c \)

Explanation:
1. 3^(-log₉x²) = 3^(-log₃|x|) = 1/|x|.
2. So the integrand is 1/|x|, whose integral is log|x|.
3. Differentiating this result returns the integrand, so option (b) 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) \( \frac{2 \left(\tan^{2}{\left(x \right)} + 5\right) \sqrt{\tan{\left(x \right)}}}{5} + c \)

Explanation:
1. Divide by cos⁴x inside: the integrand is sec⁴x/√tan x = (1 + tan²x) sec²x/√tan x.
2. Put tan x = t: ∫(1 + t²) t^(-1/2) dt = 2t^(1/2) + (2/5)t^(5/2).
3. Result = (2/5)√(tan x)(5 + tan²x).
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) \( \sin^{-1}{\left(\tan{\left(x \right)} \right)} + c \)

Explanation:
1. Write cos 2x = cos²x(1 - tan²x), so the integrand is sec²x/√(1 - tan²x).
2. Put tan x = t: ∫dt/√(1 - t²) = sin⁻¹t.
3. Result = sin⁻¹(tan 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) \( - x + 2 \tan{\left(x \right)} - 2 \sec{\left(x \right)} + c \)

Explanation:
1. Multiply numerator and denominator by (1 - sin x): the integrand is (1 - sin x)²/cos²x = sec²x - 2 sec x tan x + tan²x.
2. Using tan²x = sec²x - 1, this is 2 sec²x - 2 sec x tan x - 1.
3. Integral = 2 tan x - 2 sec x - x.
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) \( x \left(\log{\left(x \right)}^{2} - 2 \log{\left(x \right)} + 2\right) + c \)

Explanation:
1. Write ∫1·(log x)² dx and integrate by parts with u = (log x)².
2. ∫(log x)² dx = x(log x)² - 2∫log x dx = x(log x)² - 2(x log x - x).
3. Result = x[(log x)² - 2 log 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: (c) \( \left(3 x^{\frac{2}{3}} - 6 \sqrt[3]{x} + 6\right) e^{\sqrt[3]{x}} + c \)

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

Explanation:
1. Multiply by (1 - sin ax)/(1 - sin ax): the integrand is (1 - sin ax)/cos²ax = sec²ax - sec ax tan ax.
2. Integral = (1/a)(tan ax - sec ax).
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: (a) \( \frac{e^{x}}{\left(x + 6\right)^{2}} + c \)

Explanation:
1. Write (x + 4)/(x + 6)³ = 1/(x + 6)² - 2/(x + 6)³, which is f + f' with f = 1/(x + 6)².
2. So the integral of e^x(f + f') is e^x f.
3. Result = e^x/(x + 6)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: (b) \( \log{\left(\left|{x}\right| \right)} + 2 \log{\left(\left|{\sec{\left(\frac{x}{2} \right)}}\right| \right)} + c \)

Explanation:
1. Split: (1 + cos x)/(x(1 + cos x)) + x sin x/(x(1 + cos x)) = 1/x + sin x/(1 + cos x).
2. sin x/(1 + cos x) = tan(x/2), with integral -2 log|cos(x/2)| = 2 log|sec(x/2)|.
3. Result = log|x| + 2 log|sec(x/2)|.
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) \( e^{x} \csc{\left(x \right)} + c \)

Explanation:
1. Take f = cosec x; then f' = -cosec x cot x.
2. The integrand is e^x(f + f').
3. Result = e^x cosec 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: (d) \( \frac{\sin^{-1}{\left(x^{2} \right)}^{2}}{4} + c \)

Explanation:
1. Put sin⁻¹(x²) = t, so dt = 2x dx/√(1 - x⁴).
2. The integral is (1/2)∫t dt = t²/4.
3. Result = [sin⁻¹(x²)]²/4.
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: (b) \( \left(x^{x}\right)^{x} + c \)

Explanation:
1. (x^x)^x = x^(x²).
2. d/dx(x^(x²)) = x^(x²)(2x log x + x), which is the integrand.
3. Result = (x^x)^x.
4. Differentiating this result returns the integrand, so option (b) 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: (d) \( x e^{x + \frac{1}{x}} + c \)

Explanation:
1. Note d/dx(x e^(x + 1/x)) = e^(x + 1/x) + x e^(x + 1/x)(1 - 1/x²) = e^(x + 1/x)(1 + x - 1/x).
2. So the integral is x e^(x + 1/x).
3. Differentiating this result returns the integrand, so option (d) 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) \( e^{x} \log{\left(x \right)} + c \)

Explanation:
1. Here f = log x and f' = 1/x, so the integrand is e^x(f + f').
2. Result = e^x log x.
3. 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) \( - x + \log{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)} \sin{\left(x \right)} + c \)

Explanation:
1. By parts with u = log tan(x/2) and v' = cos x: du = dx/sin x.
2. ∫ = sin x log tan(x/2) - ∫(sin x/sin x) dx.
3. Result = sin x log|tan(x/2)| - x.
4. Differentiating this result returns the integrand, so option (d) 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) \( \frac14 \)

Explanation:
1. Put tan x = t, so sec²x dx = dt and the integral is ∫dt/(1 - 4t²).
2. ∫dt/(1 - 4t²) = (1/4) log|(1 + 2t)/(1 - 2t)|.
3. So k = 1/4.
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.