Indefinite Integration Questions with Solutions

12 questions2 easy9 medium1 hard

Indefinite integration is the reverse of differentiation: it finds a function whose derivative is the given function, plus an arbitrary constant C. This topic covers standard integrals, integration by substitution, integration by parts, integration using partial fractions, and integrals of special forms such as 1/(x^2 + a^2) and 1/sqrt(a^2 - x^2). Choosing the right method is the main skill. Look for a function whose derivative also appears in the integrand (substitution), a product of two different types of functions (integration by parts, choosing u with the ILATE rule), or a rational function that can be split (partial fractions). In multiple-choice questions, differentiating the answer options to check which one gives the integrand can save time. Practise the indefinite integration questions below, each with a step-by-step solution.

All Questions Easy Medium Hard

Q1.
$\int 16 - 9 x^{2} d x$ equals
A. $\frac{x}{2} 16 - 9 x^{2} + \frac{8}{3} s in^{- 1} (\frac{3 x}{4}) + C$ B. $\frac{3 x}{2} 16 - 9 x^{2} + 16 s in^{- 1} (\frac{3 x}{4}) + C$ C. $\frac{π}{2} s in^{- 1} (\frac{3 x}{4}) + \frac{9 x}{2} + C$ D. None of the above
mediumAsked in AMU MCA 2016View solution ›
Q2.
If $\int f (x) d x = g (x)$ , then $\int x^{5} f (x^{3}) d x$ is equal to
A. $\frac{1}{3} x^{3} g (x^{3}) - 3 \int x^{4} g (x^{3}) d x + C$ B. $\frac{1}{3} x^{3} g (x^{3}) - \int x^{2} g (x^{3}) d x + C$ C. $\frac{1}{3} x^{3} g (x^{3}) - \int x^{3} g (x^{3}) d x + C$ D. None of these
mediumAsked in NIMCET 2023View solution ›
Q3.
If $\int x s in x sec^{3} x d x = \frac{1}{2} [f (x) sec^{2} x + g (x) (\frac{t an x}{x})] + C$ , then which of the following is true?
A. $f (x) - g (x) = 0$ B. $f (x) \cdot g (x) = 0$ C. $f (x) + g (x) = 0$ D. $f (x) + g (x) = 1$
mediumAsked in NIMCET 2023View solution ›
Q4.
$\int e^{x} (s inh x + cos h x) d x$ is equal to
A. $e^{x} sec h x + C$ B. $e^{x} cos h x + C$ C. $s inh 2 x + C$ D. $cos h 2 x + C$
mediumAsked in NIMCET 2021View solution ›
Q5.
$\int 3^{3^{3^{x}}} \cdot 3^{3^{x}} \cdot 3^{x} d x$ is equal to
A. $\frac{3 ^{3^{x}} \cdot 3 ^{x}}{( l o g 3 ) ^{3}} + c$ B. $\frac{3 ^{3}}{( l o g 3 ) ^{3}} + c$ C. $\frac{3 ^{3^{x}}}{( l o g 3 ) ^{3}} + c$ D. $\frac{3 ^{3^{3^{x}}}}{( l o g 3 ) ^{3}} + c$
mediumAsked in NIMCET 2021View solution ›
Q6.
The value of $\int x e^{x} d x$ is equal to:
A. $2 x - e^{x - 4 x e^{x}} + C$ B. $(2 x - 4 x + 4) e^{x} + C$ C. $(2 x + 4 x + 4) e^{x} + C$ D. $(1 - 4 x) e^{x}$
mediumAsked in NIMCET 2014View solution ›
Q7.
The value of $\int \frac{x + 1}{x ( x e ^{x} + 1 )} d x$ is equal to
A. $l o g (\frac{1 + x e ^{x}}{x e ^{x}}) + C$ B. $l o g [x e^{x} (1 + x e^{x})] + C$ C. $l o g (\frac{1}{1 + x e ^{x}}) + C$ D. $l o g (\frac{x e ^{x}}{1 + x e ^{x}}) + C$
mediumAsked in NIMCET 2014View solution ›
Q8.
If $\int e^{x} (f (x) - f' (x)) d x = ϕ (x)$ , then the value of $\int e^{x} f (x) d x$ is
A. $ϕ (x) + e^{x} f (x)$ B. $ϕ (x) - e^{x} f (x)$ C. $\frac{1}{2} [e^{x} f (x) + ϕ (x)]$ D. $\frac{1}{2} [e^{x} f' (x) + ϕ (x)]$
mediumAsked in NIMCET 2015View solution ›
Q9.
The integral $\int 1 + 2 co t x (cosec x + co t x) d x$ , $(0 < x < \frac{π}{2})$ (where C is a constant of integration) is equal to
A. $2 l o g (s in \frac{x}{2}) + C$ B. $2 l o g (cos \frac{x}{2}) + C$ C. $4 l o g (cos \frac{x}{2}) + C$ D. $4 l o g (s in \frac{x}{2}) + C$
mediumAsked in NIMCET 2019View solution ›