TS EAPCET (Engineering) Integration by Substitution — practice questions
74 free MCQs with worked solutions. Tap any question for the answer + explanation, or practice them all in the app.
Practice TS EAPCET (Engineering) Integration by Substitution in the app →A student integrates x raised to the fifth power. Which antiderivative is correct?Which antiderivative of the reciprocal function 1/x is valid on the positive real line?Which function is an antiderivative of the exponential e raised to 3x?The integral of sec²x dx equals which of these?Integrating the quadratic 3x² + 2x + 1 term by term yields which function?Which function has derivative 1/(1 + x²)?Which inverse trigonometric function arises when 1 over the square root of 1 minus x squared is integrated?Which antiderivative corresponds to cos 5x?The integrand 1/(x² + 9) leads to which inverse tangent form?Integrating 1 over the square root of 4 minus x squared produces which result?Which function differentiates to sec x tan x?When the numerator 2x is the derivative of the denominator x² + 1, the integral takes which form?On an interval where cos x is nonzero, which function integrates tan x?Using the substitution u = x², the integral of x times e to the power x² becomes which function of x?Putting u = ln x, find the integral of ln x divided by x.Which function results from integrating the product x eˣ by parts?Integration by parts with u = x is applied to x multiplied by cos x. What is the result?Integration by parts treats ln x as ln x times 1. What antiderivative results?The integrand is eˣ times the sum of sin x and cos x. Which antiderivative is correct?For the integrand eˣ(1/x − 1/x²), which antiderivative holds?Partial fractions integrate 1/(x² − 1) when |x| > 1. The result isWhich logarithmic form gives the integral of 1 over the square root of x² + 1?The half-angle identity helps integrate sin²x. What is the antiderivative?Rewriting √x as x to the power 1/2, what is its antiderivative?The integral of sin cubed x times cos x, using u = sin x, equals which function?After u = x² + 1, the integral of x times the square root of x² + 1 becomes which function?Splitting 1/((x − 1)(x − 2)) into partial fractions for x > 2 gives which integral?Two rounds of integration by parts are needed for x squared times eˣ. What is the outcome?Taking tan⁻¹x as u and dx as dv, the integral isMultiplying and dividing by sec x + tan x helps find the antiderivative of sec x. Which is it?What is ∫ aˣ dx for a constant a > 0, a ≠ 1?Completing the square in x² + 2x + 5 helps integrate its reciprocal. The result isRepeating integration by parts on eˣ sin x brings back the original integral. The final value isSince sin²x + cos²x = 1, the integral of 1/(sin²x cos²x) simplifies to which function?What is the antiderivative of sin x divided by 1 + cos x?Substituting u = sin x in cos x/(1 + sin²x) gives which result?A student computes the area under y = x² from x = 0 to x = 1. What is the value of ∫₀¹ x² dx?Compute the area under one arch of the sine curve between 0 and π using an integral.Find the net area under cos x from the origin to π/2.Evaluate ∫ from 1 to e of dx/x.How much is the integral of eˣ taken between the limits 0 and 1?An odd function x³ is integrated over the symmetric interval from −1 to 1. The value isCompute ∫ from −2 to 2 of x² dx.Integrating sine squared over the first quadrant, from 0 to π/2, gives what value?Applying the property that replaces x with π/2 − x, find the integral of sin x over (sin x + cos x) between 0 Using integration by parts, the product x sin x is integrated from 0 to π. The value isFind the definite integral of x times eˣ between 0 and 1.What does ∫ from 0 to π/4 of sec²x dx equal?Between 0 and 1, the integral of 1 over the square root of 1 minus x squared equalsBetween x = 1 and x = 2, the linear function 2x + 1 is integrated. The result isWhat area lies under the V-shaped function |x − 1| between x = 0 and x = 2?Over a full period from 0 to 2π, the absolute value of sine has what total area?Using u = 1 + x², the function x/(1 + x²) is integrated over [0, 1]. What is the result?The natural logarithm is integrated from 1 up to e. What is the answer?A classic result gives the integral of log of sin x over the first quadrant asWriting sin³x as sin x(1 − cos²x), the integral over 0 to π/2 isWallis' reduction formula gives the integral of sin⁴x over [0, π/2] asThe improper integral of e⁻ˣ from 0 to infinity has the valueDifferentiating the accumulated integral of t² from 0 to x gives which function?If F(x) is the integral of sin t dt from 0 to x², what is F′(x)?The region under the parabola y = x² between x = 0 and x = 3 has what area?Find the area enclosed between the curves y = x and y = x².What area lies between the parabola y = 4 − x² and the x-axis?The average value of f(x) = x² over the interval [0, 3] isWhat is the integral of the product x(1 − x) taken over the unit interval?The integrand cos x/(1 + sin x) is integrated from 0 to π/2. What is the value?Integration by substitution is the integral counterpart of which differentiation rule?By the LIATE rule, which factor should be chosen as u when integrating x cos x?In the LIATE rule for choosing u, which type of function comes first in the list?The fundamental theorem of calculus shows that differentiation and integration areIf F is an antiderivative of a continuous f, then the integral of f from a to b equalsIn ∫₀² x(x² + 1)³ dx, the substitution u = x² + 1 changes the limits toUsing u = x² + 1, the integral of x(x² + 1)³ between x = 0 and x = 2 evaluates toHow can the result of an integration by substitution be checked?