Choose the correct answer in the given question- x2 ex3 d x- a -frac 13 e - x - 3- Cb - frac 13 e - x - 2- Cllc - frac 12 e - x - 3- C d frac -12 e - x - 2- Cl

Let I =∫ x^2 e^x^3dx Put x^3 =t ⇒ 3x^2 =dt/dx⇒ dx =dt/3x^2 ∴ I =1/3∫ e^t dt ⇒ I =1/3e^t +C =1/3e^x^3+C Hence, correct option is (a) ...

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Standard XII

Mathematics

Implicit Differentiation

Choose the co...

Question

Choose the correct answer in the given question.
$\int x^{2} e^{x^{3}} d x .$
$(a) \frac{1}{3} e^{x^{3}} + C (b) \frac{1}{3} e^{x^{2}} + C (c) \frac{1}{2} e^{x^{3}} + C (d) \frac{1}{2} e^{x^{2}} + C$

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Solution

Let $I = \int x^{2} e^{x^{3}} d x$
Put $x^{3} = t \Rightarrow 3 x^{2} = \frac{d t}{d x} \Rightarrow d x = \frac{d t}{3 x^{2}}$
$∴ I = \frac{1}{3} \int e^{t} d t \Rightarrow I = \frac{1}{3} e^{t} + C = \frac{1}{3} e^{x^{3}} + C$
Hence, correct option is (a)

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Similar questions

Choose the correct answer in each of the question.
$\int \frac{d x}{x (x^{2} + 1)}$ equals
$(a) l o g | x | - \frac{1}{2} l o g (x^{2} + 1) + C (b) l o g | x | + \frac{1}{2} l o g (x^{2} + 1) + C (c) - l o g | x | + \frac{1}{2} l o g (x^{2} + 1) + C (d) \frac{1}{2} l o g | x | + l o g (x^{2} + 1) + C$

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$(a) \frac{x}{2} \sqrt{1 + x^{2}} + \frac{1}{2} l o g ∣ ∣ x + \sqrt{1 + x^{2}} ∣ ∣ + C (b) \frac{2}{3} (1 + x^{2})^{\frac{3}{2}} + C (c) \frac{2}{3} x (1 + x^{2})^{\frac{3}{2}} + C (d) \frac{x^{2}}{2} \sqrt{1 + x^{2}} + \frac{1}{2} x^{2} l o g | x + \sqrt{1 + x^{2}} |$

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The anti-derivative of $(\sqrt{x} + \frac{1}{\sqrt{x}})$ equals
$(a) \frac{1}{3} x^{\frac{1}{3}} + 2 x^{\frac{1}{2}} + C (b) \frac{2}{3} x^{\frac{2}{3}} + \frac{1}{2} x^{2} + C (c) \frac{2}{3} x^{\frac{3}{2}} + 2 x^{\frac{1}{2}} + C (d) \frac{3}{2} x^{\frac{3}{2}} + \frac{1}{2} x^{\frac{1}{2}} + C$

\int \frac{d x}{s i n x + s i n 2 x} =

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Choose the correct answer in the given question. ∫x2ex3dx. (a)13ex3+C(b)13ex2+C(c)12ex3+C(d)12ex2+C

Let I=∫x2ex3dx Put x3=t⇒3x2=dtdx⇒dx=dt3x2 ∴I=13∫etdt⇒I=13et+C=13ex3+C Hence, correct option is (a)

Choose the correct answer in the given question.
$\int x^{2} e^{x^{3}} d x .$
$(a) \frac{1}{3} e^{x^{3}} + C (b) \frac{1}{3} e^{x^{2}} + C (c) \frac{1}{2} e^{x^{3}} + C (d) \frac{1}{2} e^{x^{2}} + C$

Let $I = \int x^{2} e^{x^{3}} d x$
Put $x^{3} = t \Rightarrow 3 x^{2} = \frac{d t}{d x} \Rightarrow d x = \frac{d t}{3 x^{2}}$
$∴ I = \frac{1}{3} \int e^{t} d t \Rightarrow I = \frac{1}{3} e^{t} + C = \frac{1}{3} e^{x^{3}} + C$
Hence, correct option is (a)