More practice questions

5. Q5. Find ∫(ax² + bx + c) dx and ∫(2x² + e^x) dx (NCERT Exercise 7.1, Q8 and Q9, page 234).

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   ax³/3 + bx²/2 + cx + C, and (2/3)x³ + e^x + C.

6. Q6. Find ∫(√x − 1/√x)² dx (NCERT Exercise 7.1, Q10, page 234).

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   The square is x − 2 + 1/x, so the integral is x²/2 − 2x + log |x| + C.

7. Q7. Find ∫(x³ − x² + x − 1)/(x − 1) dx (NCERT Exercise 7.1, Q13, page 234).

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   The top factorises as (x − 1)(x² + 1), so the integrand is x² + 1 (for x ≠ 1) and the integral is x³/3 + x + C.

8. Q8. Find ∫(1 − x)√x dx and ∫√x(3x² + 2x + 3) dx (NCERT Exercise 7.1, Q14 and Q15, pages 234–235).

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   ∫(x^1/2 − x^3/2) dx = (2/3)x^3/2 − (2/5)x^5/2 + C. ∫(3x^5/2 + 2x^3/2 + 3x^1/2) dx = (6/7)x^7/2 + (4/5)x^5/2 + 2x^3/2 + C.

9. Q9. Find ∫(2x − 3 cos x + e^x) dx and ∫(2x² − 3 sin x + 5√x) dx (NCERT Exercise 7.1, Q16 and Q17, page 235).

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   x² − 3 sin x + e^x + C, and (2/3)x³ + 3 cos x + (10/3)x^3/2 + C.

10. Q10. Find the curve y = F(x) through (1, 2) whose slope at every point is 3x² + 1.

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    F(x) = x³ + x + C, and F(1) = 2 gives 1 + 1 + C = 2, so C = 0: y = x³ + x.

11. Q11. Find F with F′(x) = 1/x for x ≠ 0, F(1) = 0 and F(−1) = 2.

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    Beyond the NCERT text: x ≠ 0 is two intervals, and each takes its own constant. F(x) = log x for x > 0 and F(x) = log (−x) + 2 for x < 0; no single log |x| + C meets both conditions.