Integrals

\(\int x^{2}\,dx\) equals:

1. a\(\dfrac{x^{2}}{2} + C\)
2. b\(\dfrac{x^{3}}{3} + C\)
3. c\(2x + C\)
4. d\(x^{3} + C\)

\(\int \sin x\,dx\) equals:

1. a\(\cos x + C\)
2. b\(-\sin x + C\)
3. c\(\sin x + C\)
4. d\(-\cos x + C\)

\(\int e^{2x}\,dx\) equals:

1. a\(e^{2x} + C\)
2. b\(2e^{2x} + C\)
3. c\(\dfrac{e^{2x}}{2} + C\)
4. d\(\dfrac{e^{x}}{2} + C\)

\(\int \dfrac{dx}{1+x^{2}}\) equals:

1. a\(\log(1+x^{2}) + C\)
2. b\(\sin^{-1} x + C\)
3. c\(\sec^{-1} x + C\)
4. d\(\tan^{-1} x + C\)

\(\int \dfrac{dx}{x\log x}\) equals:

1. a\(\dfrac{1}{\log x} + C\)
2. b\(\dfrac{(\log x)^{2}}{2} + C\)
3. c\(\log|\log x| + C\)
4. d\(x\log x + C\)

\(\int_{0}^{1} x^{2}\,dx\) equals:

1. a\(\dfrac{1}{2}\)
2. b\(\dfrac{1}{3}\)
3. c\(1\)
4. d\(\dfrac{2}{3}\)

\(\int_{0}^{\pi/2} \sin^{2} x\,dx\) equals:

1. a\(\pi\)
2. b\(\dfrac{\pi}{2}\)
3. c\(\dfrac{\pi}{4}\)
4. d\(\dfrac{1}{2}\)

\(\int x e^{x}\,dx\) equals:

1. a\(xe^{x} + e^{x} + C\)
2. b\(xe^{x} - e^{x} + C\)
3. c\(e^{x}(x+1) + C\)
4. d\(\dfrac{x^{2} e^{x}}{2} + C\)

\(\int \dfrac{2x}{x^{2}+1}\,dx\) equals:

1. a\(\dfrac{1}{x^{2}+1} + C\)
2. b\(\tan^{-1}(x^{2}+1) + C\)
3. c\(2\log x + C\)
4. d\(\log(x^{2}+1) + C\)

If \(\int f(x)\,dx = F(x) + C\), then \(\int f(2x)\,dx\) equals:

1. a\(F(2x) + C\)
2. b\(\dfrac{1}{2}F(2x) + C\)
3. c\(2F(2x) + C\)
4. d\(F\left(\dfrac{x}{2}\right) + C\)

Assertion (A): \(\int_{-a}^{a} f(x)\,dx = 0\) if \(f\) is an odd function.

Reason (R): For odd \(f\) continuous on \([-a, a]\), the substitution \(x \to -x\) shows \(\int_{-a}^{0} f = -\int_{0}^{a} f\), so the sum vanishes.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Assertion (A): \(\int_{0}^{\pi/2} \sin x\,dx = \int_{0}^{\pi/2} \cos x\,dx\).

Reason (R): Substitute \(x \to \pi/2 - x\) in either integral; \(\sin\) and \(\cos\) swap, while limits map onto each other.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Assertion (A): \(\int \dfrac{1}{\sqrt{1-x^{2}}}\,dx = \sin^{-1} x + C\).

Reason (R): \(\dfrac{d}{dx}\sin^{-1} x = \dfrac{1}{\sqrt{1-x^{2}}}\) for \(|x| < 1\).

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

1. I. \(\int (f(x) + g(x))\,dx = \int f(x)\,dx + \int g(x)\,dx\).
2. II. \(\int kf(x)\,dx = k\int f(x)\,dx\) for any constant \(k\).

Which integration statements are correct?

1. aI only
2. bII only
3. cBoth
4. dNeither

1. I. \(\int_{a}^{b} f(x)\,dx = -\int_{b}^{a} f(x)\,dx\).
2. II. \(\int (f(x)g(x))\,dx = \left(\int f(x)\,dx\right)\left(\int g(x)\,dx\right)\).

Which statements are correct?

1. aI only
2. bII only
3. cBoth
4. dNeither

1. I. \(\int_{0}^{a} f(x)\,dx = \int_{0}^{a} f(a-x)\,dx\).
2. II. If \(f\) is even, \(\int_{-a}^{a} f(x)\,dx = 2\int_{0}^{a} f(x)\,dx\).

Which statements are correct?

1. aI only
2. bII only
3. cNeither
4. dBoth

Match each integral with its result:

1. a(i)-S, (ii)-R, (iii)-Q, (iv)-P
2. b(i)-Q, (ii)-P, (iii)-S, (iv)-R
3. c(i)-P, (ii)-Q, (iii)-R, (iv)-S
4. d(i)-R, (ii)-S, (iii)-P, (iv)-Q

Match each integral with its value:

1. a(i)-P, (ii)-Q, (iii)-R, (iv)-S
2. b(i)-Q, (ii)-P, (iii)-S, (iv)-R
3. c(i)-S, (ii)-R, (iii)-Q, (iv)-P
4. d(i)-R, (ii)-S, (iii)-P, (iv)-Q

Position \(s(t) = \int v(t)\,dt\) where \(v(t) = 2t + 3\) m/s and \(s(0) = 5\) m. Then \(s(2)\) equals:

1. a11 m
2. b15 m
3. c13 m
4. d17 m

\(\int_{0}^{\pi} \dfrac{x\sin x}{1+\cos^{2} x}\,dx\) equals:

1. a\(\pi\)
2. b\(\dfrac{\pi}{4}\)
3. c\(\pi^{2}\)
4. d\(\dfrac{\pi^{2}}{4}\)