Find the anti derivative of sin 2x.
Hint. Since d/dx(cos2x) = -2sin2x, guess a cos2x term and fix the constant.
We want F with F'(x)=sin2x. Since d/dx(cos2x)=-2sin2x, we take F(x)=-(1/2)cos2x, so that F'(x)=-(1/2)(-2sin2x)=sin2x.
✦ -(1/2)cos 2x + C
Find the anti derivative of cos 3x.
Hint. d/dx(sin3x)=3cos3x, so scale by 1/3.
Since d/dx[(1/3)sin3x] = (1/3)(3cos3x) = cos3x, this is the anti derivative we need.
✦ (1/3)sin 3x + C
Find the anti derivative of e^(2x).
Hint. d/dx(e^(2x))=2e^(2x), so scale by 1/2.
Because d/dx[(1/2)e^(2x)] = (1/2)(2e^(2x)) = e^(2x), the required anti derivative follows directly.
✦ (1/2)e^(2x) + C
Find the anti derivative of (ax+b)^2.
Hint. Use the substitution rule for a linear inner function: integrating u^2 gives u^3/3, then divide by the derivative of the inner function.
Treating (ax+b) as a single variable u with du=a dx, we get integral (1/a)u^2/3a = (ax+b)^3/(3a), since differentiating (ax+b)^3/(3a) gives 3a(ax+b)^2/(3a) = (ax+b)^2.
✦ (ax+b)^3 / (3a) + C, a not 0
Find the anti derivative of sin 2x - 4e^(3x).
Hint. Integrate term by term, since integration is linear.
Splitting the sum, integral of sin2x is -(1/2)cos2x, and integral of -4e^(3x) is -(4/3)e^(3x); adding these gives the result.
✦ -(1/2)cos 2x - (4/3)e^(3x) + C
Find: integral of (4e^(3x)+1) dx.
Hint. Integrate the exponential and constant terms separately.
The integral of 4e^(3x) is (4/3)e^(3x), and the integral of the constant 1 is x, so adding these two pieces gives the answer.
✦ (4/3)e^(3x) + x + C
Find: integral of x^2(1 - 1/x^2) dx.
Hint. Expand the product first — it simplifies to a plain polynomial.
Expanding, x^2(1-1/x^2) = x^2-1, which integrates term by term using the power rule.
✦ x^3/3 - x + C
Find: integral of (ax^2+bx+c) dx.
Hint. Apply the power rule x^n -> x^(n+1)/(n+1) to each term.
Integrating term by term using the power rule on each power of x gives the antiderivative directly.
✦ a x^3/3 + b x^2/2 + c x + C
Find: integral of (2x^2+e^x) dx.
Hint. Integrate the polynomial and exponential parts separately.
The integral of 2x^2 is (2/3)x^3 by the power rule, and the integral of e^x is e^x itself, since e^x is its own antiderivative.
✦ (2/3)x^3 + e^x + C
Find: integral of (sqrt(x) - 1/sqrt(x))^2 dx.
Hint. Expand the square before integrating — the cross term simplifies nicely.
Expanding, (sqrt(x)-1/sqrt(x))^2 = x - 2 + 1/x, because the middle term is -2sqrt(x)(1/sqrt(x))=-2. Integrating term by term gives the result, using integral of 1/x = log|x|.
✦ x^2/2 - 2x + log|x| + C
Find: integral of (x^3+5x^2-4)/x^2 dx.
Hint. Divide each term in the numerator by x^2 first, so the fraction splits into simple powers of x.
Dividing through, (x^3+5x^2-4)/x^2 = x+5-4/x^2, and integrating term by term (using integral of x^-2 = -x^-1) gives the answer.
✦ x^2/2 + 5x + 4/x + C
Find: integral of (x^3+3x+4)/sqrt(x) dx.
Hint. Divide each numerator term by x^(1/2) to reduce to a sum of pure powers of x.
Dividing through by x^(1/2) gives x^(5/2)+3x^(1/2)+4x^(-1/2), so applying the power rule to each of the three terms and adding gives the final antiderivative.
✦ (2/7)x^(7/2) + 2x^(3/2) + 8x^(1/2) + C
Find: integral of (x^3-x^2+x-1)/(x-1) dx.
Hint. Factor the numerator by grouping — it has (x-1) as a factor, so the fraction simplifies before you integrate.
Grouping, x^3-x^2+x-1 = x^2(x-1)+1(x-1) = (x-1)(x^2+1), so the fraction reduces to x^2+1 once (x-1) cancels, which integrates term by term.
✦ x^3/3 + x + C
Find: integral of (1-x)*sqrt(x) dx.
Hint. Distribute sqrt(x) across the bracket first.
Expanding, (1-x)sqrt(x) = x^(1/2) - x^(3/2), so integrating each power of x separately using the power rule gives the result.
✦ (2/3)x^(3/2) - (2/5)x^(5/2) + C
Find: integral of sqrt(x)*(3x^2+2x+3) dx.
Hint. Multiply x^(1/2) into every term of the bracket, then integrate each power.
Distributing, sqrt(x)(3x^2+2x+3) = 3x^(5/2)+2x^(3/2)+3x^(1/2), and integrating each term with the power rule gives the antiderivative.
✦ (6/7)x^(7/2) + (4/5)x^(5/2) + 2x^(3/2) + C
Find: integral of (2x-3cos x+e^x) dx.
Hint. Integrate each of the three terms independently.
Term by term: integral of 2x is x^2, integral of -3cosx is -3sinx, and integral of e^x is e^x, so summing these three gives the answer.
✦ x^2 - 3sin x + e^x + C
Find: integral of (2x^2-3sin x+5sqrt(x)) dx.
Hint. Integrate the polynomial, trigonometric, and root terms one at a time.
Term by term: integral of 2x^2 is (2/3)x^3, integral of -3sinx is 3cosx, and integral of 5x^(1/2) is 5*(2/3)x^(3/2)=(10/3)x^(3/2); adding these gives the result.
✦ (2/3)x^3 + 3cos x + (10/3)x^(3/2) + C
Find: integral of sec x(sec x + tan x) dx.
Hint. Expand the product — both resulting terms are standard derivatives you already know.
Expanding gives sec^2x+secx tanx, and since d/dx(tanx)=sec^2x and d/dx(secx)=secx tanx, integrating term by term gives the answer directly.
✦ tan x + sec x + C
Find: integral of (sec^2 x)/(cosec^2 x) dx.
Hint. Rewrite cosec^2 x as 1/sin^2 x, so the integrand becomes sec^2x . sin^2x, which simplifies using sin/cos identities.
Since 1/cosec^2x = sin^2x, the integrand becomes sec^2x sin^2x = sin^2x/cos^2x = tan^2x. Using tan^2x = sec^2x-1, the integral splits into integral of sec^2x minus integral of 1.
✦ tan x - x + C
Find: integral of (2-3sin x)/cos^2 x dx.
Hint. Split the fraction into two separate terms first, both standard forms once divided by cos^2 x.
Splitting, (2-3sinx)/cos^2x = 2sec^2x - 3(sinx/cos^2x) = 2sec^2x - 3secx tanx, since sinx/cos^2x = (1/cosx)(sinx/cosx) = secx tanx. Integrating gives the result.
✦ 2tan x - 3sec x + C
Choose the correct answer: the anti derivative of (sqrt(x)+1/sqrt(x)) equals (A) (1/3)x^(1/3)+2x^(1/2)+C (B) (2/3)x^(2/3)+(1/2)x^2+C (C) (2/3)x^(3/2)+2x^(1/2)+C (D) (3/2)x^(3/2)+(1/2)x^2+C
Hint. Integrate x^(1/2) and x^(-1/2) separately using the power rule, then match against the options.
Integral of x^(1/2) is (2/3)x^(3/2), and integral of x^(-1/2) is 2x^(1/2), since (1/2)+1=3/2 and -(1/2)+1=1/2 respectively. Adding these matches option (C).
✦ (C) (2/3)x^(3/2) + 2x^(1/2) + C
Choose the correct answer: if d/dx f(x) = 4x^3-3/x^4 such that f(2)=0, then f(x) is (A) x^4+1/x^3-129/8 (B) x^3+1/x^4+129/8 (C) x^4+1/x^3+129/8 (D) x^3+1/x^4-129/8
Hint. Integrate 4x^3-3x^-4 to get a general f(x)+C, then use f(2)=0 to pin down C.
Integrating, f(x) = x^4 + x^-3 + C = x^4+1/x^3+C. Since f(2)=0, we get 16+1/8+C=0, so C=-129/8. This matches option (A), because the sign on the constant is negative.
✦ (A) x^4 + 1/x^3 - 129/8
Solutions written by the tuition.in editorial team and checked against lemh201.pdf (NCERT, Reprint 2026-27). Questions are referenced from the NCERT textbook for identification.
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